Influence of cumulative damage on seismic response modification factors of elastic perfectly plastic oscillators

Abstract

Based upon elastic perfectly plastic oscillators, this research investigates the influence of cumulative damage on seismic response modification factors. To take into account the cumulative damage due to hysteretic energy absorption, the Park–Ang damage index was adopted as the damage measure. A large number of iterative nonlinear response history analyses were conducted for the oscillators with periods and other features representative of conventional civil structures. Analysis results show that a significant portion of the total damage in a system reaching its damage limit state is due to hysteretic energy absorption. Consequently, the maximum tolerable ductility demand in the system under earthquake loading is lower than its ductility capacity determined under the monotonic loading condition. Moreover, the existing formulae for calculating response modification factors are shown to be inadequate to capture the effect of cumulative damage. A new method was developed for improved estimates of the response modification factors.

Keywords

cumulative damage ductility hysteretic energy Park–Ang damage index response modification factors seismic demands

Introduction

Modern seismic design codes such as ASCE/SEI 7-10 (2010) allow designing a civil structure for a base shear demand considerably lower than that associated with pure elastic response of the system under an earthquake event of expected level. Specifically, the elastic base shear demand can be determined through the use of linear elastic response spectrum of the construction site; the base shear demand actually used in proportioning structural members of the system is commonly reduced from the elastic base shear demand by the response modification factor, which is also known as the R factor in the field of earthquake engineering (ASCE/SEI 7-10, 2010). The response modification factors for conventional lateral force resisting systems can be found in many seismic design provisions such as ASCE/SEI 7-10 (2010). Note that different lateral force resisting systems may have different response modification factors.

Compared with the design considering elastic base shear demand, the design based on reduced base shear demand can be more economical (and even more practical in some cases). However, the base shear assumed in design, if overly reduced, can result in excessive inelastic structural behaviors (i.e. significant structural damage), leading to unfavorable performance of the system during severe earthquake events. A reduced base shear can be only used in seismic design if the resulting structural damage is controlled to a tolerable level. As such, a good estimation of the response modification factor is important to achieve the expected seismic performance in a system. Additionally, the improvement of response modification factors has been identified as a way to improve the adequacy of seismic design provisions (Bertero et al., 1991; Porush, 1988).

As introduced in detail in a following section, most of the existing methods to determine the response modification factors were proposed with the intent of limiting the structural damage due to the maximum displacement response of a system under earthquake loading. However, it is recognized that severe earthquakes impose not only large displacement demand but also numerous inelastic loading reversals on a system. Consequently, the system may have to absorb a significant amount of hysteretic energy, resulting in cumulative structural damage (e.g. low-cycle fatigue) in the system. Compared with the research focusing on the damage due to the maximum system displacement response in determination of the response modification factors, the work considering the cumulative damage due to progressively dissipated hysteretic energy is very limited. Crucial knowledge gaps exist in understanding and quantifying the effect of cumulative damage on response modification factors. It is an impediment to the widespread acceptance of the current seismic design provisions for mitigating the catastrophic effects of earthquakes.

As a logical first step, this research focuses on analyses of simple oscillators to examine the influence of cumulative damage in determination of response modification factors. Note that recent research efforts have been made to estimate the response modification factors through the more sophisticated incremental dynamic analyses (IDA) of archetypes of representative building systems (Federal Emergency Management Agency (FEMA), 2009); however, such a procedure requires extensive modeling efforts, is computationally expensive, and may be too onerous to parametrically illustrate the influence of cumulative damage on response modification factors. Therefore, this research was conducted as a pilot study with the purpose of generating the key knowledge needed for forming the basis for future in-depth research in the field.

The objectives of this investigation are (1) to generate a database of response modification factors based upon a proper damage measure that combines the damages caused by both the maximum displacement response and the hysteretic energy dissipation; (2) to compare the damages, respectively, contributed by the maximum displacement response and hysteretic energy dissipation in a system reaching its damage limit state; (3) to identify the limitations of the existing formulae for calculating response modification factors; and (4) to develop a practical method for improved estimates of response modification factors.

This research first revisits the existing method to determine oscillator response modification factors and then explains why the cumulative damage should be considered, followed by the simulation work conducted (including adoption of the unified damage index, selection of the ground motion inputs, assumption of the oscillator hysteretic behavior, and iteration scheme) and result discussions (including comparisons of the response modification factors determined with and without consideration of cumulative damage and development of a method for improved estimates of response modification factors).

Existing method to determine response modification factors

Although the response modification factors specified in current building codes are primarily based on observation of the performance of different structural systems in past strong earthquakes (Newmark and Hall, 1973), the more rational method to determine the response modification factors exists and is briefly described below based on elastic perfectly plastic (EPP) oscillators. The response history of an EPP oscillator subjected to earthquake ground motions is given by the solution of the following equation of motion

m \overset{\cdot\cdot}{u} (t) + c \overset{\cdot}{u} (t) + F (t) = - m {\overset{\cdot\cdot}{u}}_{g} (t)

(1)

where m, c, F(t), and u(t) represent the mass, damping coefficient, restoring force, and relative displacement response of the oscillator, respectively; u_g(t) represents the ground displacement history; and the dot over a displacement quantity represents its derivative with respect to time.

The maximum inelasticity experienced by the oscillator under a specific ground motion is quantified by the ductility factor, µ, which is defined as

μ = \frac{max | u (t) |}{u_{y}}

(2)

where u_y is yield displacement of the oscillator. Based on equation (2), µ is larger than 1.0 if the oscillator responds inelastically, while it is less than or equal to 1.0 if the oscillator remains elastic.

To limit structural damage caused by the maximum displacement response, the response modification factor, R, can be computed as

R (μ_{o}) = \frac{F_{y} (μ = 1)}{F_{y} (μ = μ_{o})}

(3)

where F_y(µ = 1) represents the lowest strength required to avoid yielding behavior in the oscillator under a given ground motion and F_y(µ = µ_o) represents the strength leading to the ductility factor, µ, equal to a pre-determined target value, µ_o, when subjected to the same ground motion. Typically, µ_o is taken to be the ductility capacity of the system under the monotonic loading condition, µ_c. Note that the actual strength of a system is usually higher than its design demand due to the conservative assumptions typically made in its strength determination (e.g. ignoring material strain hardening). Technically, the response modification factor defined in equation (3) should take into account the overstrength in a system. However, this investigation neglected oscillator overstrength for simplicity, making the response modification factors determined in equation (3) the same as the strength reduction factors defined in past research (Miranda and Bertero, 1994). It is recognized that equation (3) underestimates the response modification factor of a system when overstrength in the system is neglected.

For a specific ground motion, determination of F_y(µ = µ_o) involves iteration of the oscillator strength F_y in equation of motion, that is, equation (1), until the computed ductility demand, µ, is, within a certain tolerance, the same as the target ductility µ_o. Generally, the ductility demand, µ, associated with a given earthquake ground motion increases when strength of the oscillator decreases as illustrated in Figure 1. Note in Figure 1 that strength of the system is normalized by reactive weight of the oscillator, w. Therefore, according to the above-mentioned definition, the system with a better ductility capacity, µ_c, can be designed for a lower strength demand.

Figure 1.

Strength required to maintain a constant ductility.

Why cumulative damage should be considered

Although the response modification factors determined according to equation (3) helps limit the structural damage caused by the maximum displacement response (i.e. ductility), it should be recognized that the level of structural damage does not depend solely on ductility. The cumulative damage resulting from gradually dissipated hysteretic energy through numerous inelastic cyclic loading reversals also affects the total damage in the system. To better understand the influence of cumulative damage due to hysteretic energy absorption, multiplying du on both sides of equation (1) and integrating with respect du, one can obtain the following equation

\int_{0}^{u} m \overset{\cdot\cdot}{u} (t) du + \int_{0}^{u} c \overset{\cdot}{u} (t) du + \int_{0}^{u} F (t) du = - \int_{0}^{u} m {\overset{\cdot\cdot}{u}}_{g} (t) du

(4)

Equation (4) can be rewritten as

E_{K} + E_{D} + E_{H} + E_{S} = E_{I}

(5)

where

E_{K} = \int_{0}^{u} m \overset{\cdot\cdot}{u} (t) du

(6)

E_{D} = \int_{0}^{u} c \overset{\cdot}{u} (t) du

(7)

E_{H} + E_{S} = \int_{0}^{u} F (t) du

(8)

E_{I} = - \int_{0}^{u} m {\overset{\cdot\cdot}{u}}_{g} (t) du

(9)

Essentially, E_K, E_D, E_H, E_S, and E_I are kinetic energy, energy dissipated by viscous damping, energy dissipated by inelastic hysteretic behavior, elastic strain energy, and earthquake energy imparted to the system, respectively. Equation (5) actually represents energy balance during the ground motion duration.

When an oscillator that undergoes inelastic loading excursions comes back to rest at the end of a ground motion, the kinetic and elastic strain energy terms, that is, E_K and E_S, vanish. As a result, at the end of the ground motion, equation (5) becomes

E_{D} + E_{H} = E_{I}

(10)

Equation (10) indicates that the energy released from an earthquake into a system needs to be dissipated by inelastic hysteretic behavior and viscous damping of the system. For conventional lateral force resisting systems, E_H is typically larger and in most cases much larger than E_D, and hence it is the major source to dissipate the input earthquake energy. Therefore, similar to ductility capacity, the capacity of a system to absorb hysteretic energy (i.e. its capacity to sustain cumulative low-cycle fatigue damage) should be considered in determination of the response modification factors.

Beyond equation (10), some seismic design guidelines also highlight the importance of taking into account the hysteretic energy absorption capacity of a system in its seismic design and evaluation. For example, the 2003 NEHRP Recommended Provisions and Commentary for Seismic Regulations for New Buildings and Other Structures (FEMA, 2003) explicitly specifies that structural systems with larger hysteretic energy dissipation capacity should be assigned higher response modification factors (i.e. the R factor), resulting in design for lower forces, than systems with relatively limited hysteretic energy dissipation capacity.

To date, many simplified formulae have been proposed to quantity the R factor based on numerical simulations of representative oscillators (Hidalgo and Arias, 1990; Lai and Biggs, 1980; Miranda, 1993; Nassar and Krawinkler, 1991; Riddell et al., 1989; Vidic et al., 1992; to name a few). However, most of them are based on equation (3) which only considers the damage caused by the maximum system displacement response. Several researchers have expressed their concerns about the lack of rationality in deriving these formulae for determination of the R factors which completely ignore the cumulative damage caused by hysteretic energy absorption. The damage due to hysteretic energy absorption may be considered through equation (3) if strength and stiffness deteriorations due to low-cycle fatigue can be captured by the oscillator hysteretic models; however, prior investigators only assumed simple hysteretic behavior in their oscillators (Hidalgo and Arias, 1990; Lai and Biggs, 1980; Miranda, 1993; Nassar and Krawinkler, 1991; Riddell et al., 1989; Vidic et al., 1992). Therefore, there is a significant knowledge gap in the field to better understand the effect of cumulative damage on response modification factors.

Unified damage measure

As discussed in section “Existing method to determine response modification factors,” determination of the response modification factors requires a unified damage measure combining the damage contributions of both the maximum displacement response and the hysteretic energy absorption. One popular example of such damage measures is the well-known Park–Ang damage index (Park and Ang, 1985) defined below

DI = \frac{μ}{μ_{c}} + β \frac{E_{H}}{F_{y} u_{y} μ_{c}}

(11)

where DI represents the Park–Ang damage index, β is a non-dimensional parameter considering the effect of cyclic loading on cumulative damage which can be determined experimentally, and the other variables have been defined previously. Technically, β correlates with the capacity of a system to absorb hysteretic energy, and it is lower in a system with better hysteretic energy dissipation capacity. For reinforce concrete structures, a β value of 0.15 was originally proposed by Park and Ang (1985). Over the past three decades, researchers have proposed and used various β values ranging from 0.025 to 0.25 for different types of structural systems (Ghosh et al., 2011). Note that the first and second terms on the right-hand side of equation (11) essentially consider the damages caused by the maximum displacement response (i.e. ductility) and hysteretic energy absorption, respectively.

It is noted that DI is a dimensionless parameter. The DI value of 0 corresponds to an undamaged state (i.e. the system remains elastic), while the DI value of 1.0 or larger indicates a limit state of fully damaged or collapsed (which should be absolutely avoided in seismic design). The intermediate values between 0 and 1.0 correspond to the partial damage levels which may be acceptable.

Excluding the contribution of recoverable deformation (i.e. elastic portion of the resultant deformation) from the first term of the Park–Ang damage index gives the following modified version of the Park–Ang damage index (Kunnath et al., 1992), which has also been used in some recent investigations (Ghosh et al., 2011; Kunnath et al., 1992; Samanta et al., 2012)

DI = \frac{μ - 1}{μ_{c} - 1} + β \frac{E_{H}}{F_{y} u_{y} μ_{c}}

(12)

Figure 2 compares the DI values, respectively, predicted by equations (11) and (12) for EPP oscillators with different periods and under an arbitrarily selected earthquake. As shown, predictions from equation (11) are consistently higher than those from equation (12). Given that overestimates of DI lead to smaller response modification factors and hence conservative seismic design forces, this research only focuses on the Park–Ang damage index defined in equation (11). However, the analysis work presented in the following sections can be similarly performed for the damage index defined in equation (12) if needed.

Figure 2.

Comparison of Park–Ang damage indices: equation (11) versus equation (12).

Introducing DI defined in equation (11) into equation (3) as the damage measure for controlling system performance gives

R (μ_{c}, β) = \frac{F_{y} (μ = 1)}{F_{y} (DI = \frac{μ}{μ_{c}} + β \frac{E_{H}}{F_{y} u_{y} μ_{c}} = 1)}

(13)

It is noted that equation (13) becomes the same as equation (3) when β is equal to 0.

Iterative nonlinear response history analyses

Based on equations (1) and (13), iterative nonlinear response history analyses (RHA) were conducted to generate the database of R with consideration of a broad range of other parameters including µ_c, β, and various ground motion inputs.

Ground motion input

The far-field earthquake record set recommended in the ATC 63 and FEMA P695 projects (FEMA, 2009) for seismic performance evaluation was used as ground motion inputs in RHA. The far-field record set includes 22 ground motion pairs (44 individual components) selected with a special consideration of source magnitude, source type, site conditions, site-to-source distance, and number of records per event (FEMA, 2009).

Table 1 summarizes the magnitude, year, name of the event, name of the station, fault characteristics, epicentral distance, and peak ground acceleration (PGA) of the ground motions. As shown, the 22 ground motion pairs are from 14 events that occurred between 1971 and 1999. Of the 14 events, 8 were Californian earthquakes and 6 were from five different foreign countries. Event magnitudes range from M6.5 to M7.6 with an average magnitude of M7.0. Fifteen records are from events of predominantly strike-slip faulting. The epicentral distances range from 8.7 to 98.2 km with an average of 39.6 km. To keep record-to-record variability due to inherent differences in event magnitude, distance to source, source type, and site conditions, the as-recorded ground motions were used as the RHA inputs (i.e. no modifications were made to the original ground motions). Figure 3 shows the linear elastic response spectra of the ATC/FEMA far-field earthquake set. As shown, the selected earthquakes have a wide range of spectral amplitude and spectral shape, which help generate unbiased results from RHA.

Table 1.

Summary of input earthquake records.

ID No.	Earthquake	Fault type	Epicentral distance (km)	Magnitude	Station	PGA (g)
GM1	1994 Northridge	Thrust	13.3	6.7	Beverly Hills—Mulhol	0.42
GM2	1994 Northridge	Thrust	13.3	6.7	Beverly Hills—Mulhol	0.52
GM3	1994 Northridge	Thrust	26.5	6.7	Canyon Country-WLC	0.41
GM4	1994 Northridge	Thrust	26.5	6.7	Canyon Country-WLC	0.48
GM5	1999 Duzce, Turkey	Strike-slip	41.3	7.1	Bolu	0.73
GM6	1999 Duzce, Turkey	Strike-slip	41.3	7.1	Bolu	0.82
GM7	1999 Hector Mine	Strike-slip	26.5	7.1	Hector	0.27
GM8	1999 Hector Mine	Strike-slip	26.5	7.1	Hector	0.34
GM9	1979 Imperial Valley	Strike-slip	33.7	6.5	Delta	0.24
GM10	1979 Imperial Valley	Strike-slip	33.7	6.5	Delta	0.35
GM11	1979 Imperial Valley	Strike-slip	29.4	6.5	El Centro Array #11	0.36
GM12	1979 Imperial Valley	Strike-slip	29.4	6.5	El Centro Array #11	0.38
GM13	1995 Kobe, Japan	Strike-slip	8.7	6.9	Nishi-Akashi	0.51
GM14	1995 Kobe, Japan	Strike-slip	8.7	6.9	Nishi-Akashi	0.50
GM15	1995 Kobe, Japan	Strike-slip	46	6.9	Shin-Osaka	0.24
GM16	1995 Kobe, Japan	Strike-slip	46	6.9	Shin-Osaka	0.21
GM17	1999 Kocaeli, Turkey	Strike-slip	98.2	7.5	Duzce	0.31
GM18	1999 Kocaeli, Turkey	Strike-slip	98.2	7.5	Duzce	0.36
GM19	1999 Kocaeli, Turkey	Strike-slip	53.7	7.5	Arcelik	0.22
GM20	1999 Kocaeli, Turkey	Strike-slip	53.7	7.5	Arcelik	0.15
GM21	1992 Landers	Strike-slip	86	7.3	Yermo Fire Station	0.24
GM22	1992 Landers	Strike-slip	86	7.3	Yermo Fire Station	0.15
GM23	1992 Landers	Strike-slip	82.1	7.3	Coolwater	0.28
GM24	1992 Landers	Strike-slip	82.1	7.3	Coolwater	0.42
GM25	1989 Loma Prieta	Strike-slip	9.8	6.9	Capitola	0.53
GM26	1989 Loma Prieta	Strike-slip	9.8	6.9	Capitola	0.44
GM27	1989 Loma Prieta	Strike-slip	31.4	6.9	Gilroy Array #3	0.56
GM28	1989 Loma Prieta	Strike-slip	31.4	6.9	Gilroy Array #3	0.37
GM29	1990 Manjil, Iran	Strike-slip	40.4	7.4	Abbar	0.51
GM30	1990 Manjil, Iran	Strike-slip	40.4	7.4	Abbar	0.50
GM31	1987 Superstition Hills	Strike-slip	35.8	6.5	El Centro Imp. Co.	0.36
GM32	1987 Superstition Hills	Strike-slip	35.8	6.5	El Centro Imp. Co.	0.26
GM33	1987 Superstition Hills	Strike-slip	11.2	6.5	Poe Road (temp)	0.45
GM34	1987 Superstition Hills	Strike-slip	11.2	6.5	Poe Road (temp)	0.30
GM35	1992 Cape Mendocino	Thrust	22.7	7.0	Rio Dell Overpass	0.39
GM36	1992 Cape Mendocino	Thrust	22.7	7.0	Rio Dell Overpass	0.55
GM37	1999 Chi-Chi, Taiwan	Thrust	32	7.6	CHY101	0.35
GM38	1999 Chi-Chi, Taiwan	Thrust	32	7.6	CHY101	0.44
GM39	1999 Chi-Chi, Taiwan	Thrust	77.5	7.6	TCU045	0.47
GM40	1999 Chi-Chi, Taiwan	Thrust	77.5	7.6	TCU045	0.51
GM41	1971 San Fernando	Thrust	39.5	6.6	LA—Hollywood Stor	0.21
GM42	1971 San Fernando	Thrust	39.5	6.6	LA—Hollywood Stor	0.17
GM43	1976 Friuli, Italy	Thrust	20.2	6.5	Tolmezzo	0.35
GM44	1976 Friuli, Italy	Thrust	20.2	6.5	Tolmezzo	0.31

PGA: peak ground acceleration.

Figure 3.

Linear elastic response spectra of ATC/FEMA far-field ground motions.

Oscillator hysteretic behavior

Many hysteretic models have been proposed for oscillators in past investigations to capture the nonlinear behavior of structural components, assemblies, and systems. Although the authors did recognize that deterioration of strength and stiffness will affect the results, only the EPP hysteretic behavior was considered in the oscillators of this investigation for the following reasons: (1) it is simple and has been widely used in investigations of nonlinear structural dynamics (Chopra, 2001), (2) it has been shown to be reasonably representative of nonlinear behavior of conventional structural systems (Van de Lindt and Goh, 2004), and (3) it has been widely used in past studies of response modification factors (Lai and Biggs, 1980; Newmark and Hall, 1973; Riddell and Newmark, 1979) and other aspects of earthquake engineering (Araya and Saragoni, 1980; Bertero et al., 1978; Cornell, 1997; Rahnama and Manuel, 1996; Saragoni, 1985); and adoption of this model helps generating results for direct comparisons (which will be discussed in detail in section “Result discussions”). As typically done in the field of earthquake engineering, oscillator damping ratio was assumed to be 0.05 (Chopra, 2001).

Iteration scheme

Based on the computer code developed in this investigation, iterative RHA were conducted to generate a database of the response modification factors. For each ground motion input, fundamental period of the EPP oscillator was varied from 0.1 to 3.0 s with the increment of 0.1 s. Three values of µ_c (2, 4, and 6) and four values of β (0, 0.1, 0.2, and 0.3) were considered in the simulations, respectively. Note that these values are representative in conventional lateral force resisting systems. It is also noted that the cumulative damage due to hysteretic energy dissipation is not considered in determination of the R factor when β is selected to be 0 (see equation (11) for definition of the Park–Ang damage index). However, such a case was included here: (1) to evaluate whether the R factor obtained from this evaluation can be predicted by the methods proposed by previous investigations and (2) to illustrate the effect of cumulative damage. Figure 4 shows flow chart of the iterative RHA. As shown, at each considered period, the oscillator yield strength, F_y, is gradually reduced from F_y (µ = 1) until the computed DI value reaches 1.0. In each iteration, the decrement of F_y, ΔF_y, was selected to be 0.2% of the oscillator reactive weight. When DI reaches 1.0, equation (13) was used to calculate the R factor.

Figure 4.

Flow chart of iterative RHA.

Result discussions

The results from iterative RHA form a database for evaluating the effect of cumulative damage on response modification factors. This section first evaluates the maximum tolerable ductility demand taking into account cumulative damage, followed by comparison of the response modification factors determined with and without consideration of cumulative damage, together with the development of an empirical model for improved estimates of the response modification factors.

Maximum tolerable ductility demand accounting for cumulative damage

Technically, the maximum allowable value of µ should be equal to the ductility capacity of a system, µ_c, at the limit state. However, the hysteretic energy progressively absorbed in the system causes cumulative damage, resulting in a reduced allowable value of µ. This section presents a method to compute the reduced maximum tolerable ductility demand taking into account of the effect of cumulative damage.

Based on equation (11), setting the Park–Ang damage index equal to 1.0 for a system reaching its damage limit state gives

DI = \underset{\binom{damage due}{\binom{to maximum}{displacement}}}{\underset{︸}{\frac{μ}{μ_{c}}}} + \underset{\binom{damage due to}{\binom{hystertic energy}{absorption}}}{\underset{︸}{β \frac{E_{H}}{F_{y} u_{y} μ_{c}}}} = 1.0

(14)

As explicitly shown in equation (14), the quantity, βE_H/F_y u_y µ_c, captures the cumulative damage due to hysteretic energy absorption, which has been ignored in the conventional displacement-based determination of response modification factors. To illustrate the effect of hysteretic energy absorption, the term, βE_H/F_y u_y µ_c was calculated in each of the RHA in which the Park–Ang damage index reaches 1.0. At each considered fundamental period, βE_H/F_y u_y µ_c was computed for each of the input ground motions and median of these quantities (i.e. the 50th percentile) was output as the result of interest.

Figure 5 shows the median values of βE_H/F_y u_y µ_c determined for the assumed µ_c and β values over the considered period range. It is observed in all considered cases that βE_H/F_y u_y µ_c does not appear to be dependent upon period of the oscillator, that is, this quantity remains somewhat constant for the given µ_c and β values and over the considered period range. As shown, βE_H/F_y u_y µ_c varies from 0.1 to 0.5, indicating that about 10%–50% of the total damage in a system reaching its damage limit state is due to the cumulative damage caused by hysteretic energy dissipation. With this level of contribution, the effect of hysteretic energy dissipation should not be ignored in damage quantification. In addition, βE_H/F_y u_y µ_c is larger in a system with larger values of β and µ_c. This is consistent with the fact that a more ductile system, when reaching its limit state, tends to achieve a higher degree of nonlinearity, absorbs a larger amount of hysteretic energy, and hence sustains a higher level of cumulative damage. Based on the database of βE_H/F_y u_y µ_c, a nonlinear regression analysis was carried out. The relation of this quantity versus µ_c and β was regressed by means of Table Curve 3D. As a result, the following empirical formula was adopted

\frac{β E_{H}}{F_{y} u_{y} μ_{c}} = EXP [0.59 - \frac{0.90}{\ln (μ_{c})} - \frac{0.48}{\sqrt{β}}]

(15)

Figure 5.

Damage due to hysteretic energy absorption in a system reaching its damage limit state.

Note that equation (15) is one of the simplest equations that capture the result trend and selected after testing numerous mathematical equations considered in Table Curve 3D. Additionally, the coefficient of determination of this regression is 0.965. Moreover, it should be noted that equation (15) applies only when β is larger than zero.

Substituting equation (15) into equation (14) and solving µ give the following equation

μ = {1 - EXP [0.59 - \frac{0.90}{\ln (μ_{c})} - \frac{0.48}{\sqrt{β}}]} μ_{c}

(16)

Mathematically, µ calculated from equation (16) is lower than µ_c and it represents the reduced maximum tolerable ductility demand accounting for the cumulative damage due to hysteretic energy dissipation. To better illustrate the influence of the two parameters, µ_c and β, in equation (16), a parametric study was conducted and the results are shown in Figure 6. As shown, the maximum tolerable ductility demand reduces when β increases but to a higher degree in a system with higher ductility capacity. With the considered parameters, it is observed that the maximum tolerable ductility demand, µ, varies from 50% to 90% of the ductility capacity µ_c. Taking the cases with β equal to 0.15 as an example, which was originally recommended by Park and Ang (1985) for reinforced concrete structures, the maximum tolerable ductility demand values are 1.72, 2.31, 2.92, 3.52, and 4.12 when the ductility capacity values are 2, 3, 4, 5, and 6, respectively.

Figure 6.

Reduction in maximum tolerable ductility demands considering cumulative damage due to hysteretic energy dissipation.

R factor without consideration of cumulative damage

This section focuses on the results from the special cases with β equal to zero. Note that these cases actually did not take into account the cumulative damage in determination of the R factor. They were first discussed for the following reasons: (1) formulae predicting the response modification factors for such cases are available in the literature, which help evaluate adequacy of the RHA results, and (2) response modification factors determined from these cases are necessary to illustrate the effect of cumulative damage (which will be discussed in detail in section “R Factor with Consideration of Cumulative Damage”).

For a given period and µ_c, R was computed for each of the input ground motions and median of the R values (i.e. the 50th percentile) was output to illustrate the central tendency of this parameter. The results of R are illustrated in Figure 7. As mentioned, early studies proposed different formulae to calculate the R factor. For comparison purpose, Figure 7 also includes predictions of R from some of the formulae proposed by past investigations including Newmark and Hall (1973), Lai and Biggs (1980), Riddell et al. (1989), Hidalgo and Arias (1990), Nassar and Krawinkler (1991), Vidic et al. (1992), and Miranda (1993). Note that these formulae are not reviewed in detail in this article due to the space constraint. However, they can be found in the corresponding literature and a historical review of these formulae is available elsewhere (Miranda, 1993).

Figure 7.

Comparison of response modification factors (when β = 0).

As compared in Figure 7, trends of the RHA results generally agree well with those predicted by the existing formulae. As shown, the R factor increases with fundamental period over the short period range (from 0 to 0.5 s). When fundamental period of the oscillator is longer than 0.5 s, the R factor reaches a constant value similar to µ_c. This observation is consistent with the classic R–µ_c relationship determined from the Equal Displacement Rule which can be found in many structural dynamics textbooks (Chopra, 2001). To compare respective goodness-of-fit of the previously proposed formulae, the following root-mean-square of R, denoted as R_RMS, is defined

R_{RMS} = \sqrt{\frac{\sum_{i = 1}^{N} {(R_{RHA, i} - R_{predicted, i})}^{2}}{N}}

(17)

where R_RHA,i represents the response modification factor at period T_i predicted by RHA; R_predicted,i represents the response modification factor at period T_i predicted by a formula proposed by past investigations; and N is the number of periods at which goodness-of-fit is evaluated.

Based on equation (17), a formula with better goodness-of-fit should have a smaller R_RMS value. Table 2 compares R_RMS of the formulae proposed by past investigations and their rank with respect to goodness-of-fit (i.e. rank based upon minimum R_RMS). As shown, among these considered formulae, the one proposed by Hidalgo and Arias (1990) has the best average rank and is considered to be the one best capturing the trend of response modification factors obtained from this investigation.

Table 2.

Comparison of roots of mean squares of response modification factors.

Model	µ_c = 2		µ_c = 4		µ_c = 6		Average rank
Model	R _RMS	Rank	R _RMS	Rank	R _RMS	Rank	Average rank
Newmark and Hall (1973)	0.134	4	0.400	4	0.99	4	4.00
Lai and Biggs (1980)	0.213	7	0.650	6	N/A	N/A	N/A
Riddell et al. (1989)	0.153	5	0.382	3	0.73	1	3.00
Hidalgo and Arias (1990)	0.129	2	0.345	1	0.86	2	1.67
Nassar and Krawinkler (1991)	0.105	1	0.544	5	1.56	6	3.33
Vidic et al. (1992)	0.129	3	0.350	2	0.98	3	2.33
Miranda (1993)	0.202	6	0.699	7	1.34	5	4.33

R factor with consideration of cumulative damage

The iterative RHA were then extended to include consideration of cumulative damage. The key parameter to this end is β. Technically, a system with better hysteretic absorption capacity should be assigned a lower β value since the Park–Ang damage index (see equation (11)) will be lower if a lower β value is used. For comparison purpose, β was selected to be 0.1, 0.2, and 0.3, representing the systems with excellent, moderate, and limited hysteretic energy absorption capacities. Similar to the analyses described in section “R factor without consideration of cumulative damage,” at a given period, R was calculated for a given value of µ_c using each of the input ground motions; and median (i.e. the 50th percentile) and coefficient of variation (COV) of these R values were output to illustrate central tendency and dispersion of this parameter. The results are illustrated in Figures 8 and 9.

Figure 8.

Comparison of response modification factors.

Figure 9.

Coefficient of variation of response modification factors.

As shown in Figure 8, for a given period and µ_c, R is always higher in a system with a smaller β value and it reaches the maximum when β reduces to zero (which corresponds to the cases without consideration of cumulative damage). Given that ignorance or underestimate of the cumulative damage leads to overestimates of R, which in turn leads to underestimate of base shear and accordingly unsafe seismic design, it is necessary to consider the effect of hysteretic energy absorption in determination of R. Figure 9 shows COV of R. As shown, although smaller values of β and larger values of period and µ_c seem to slightly increase the dispersion of the data, the COV results from all cases moderately scatter around 0.3, indicating that the RHA produced stable results while fairly capturing inherent aleatory (i.e. record-to-record) variability.

Empirical method for improved estimate of R

As shown in Figure 8, response modification factors can be significantly lower if cumulative damage is considered. To better capture this phenomenon, a deamplification factor, ψ, is defined as the ratio of the R factor with consideration of cumulative damage to that without consideration of cumulative damage. Technically, ψ is primarily influenced by characteristics of a system including its fundamental period, ductility capacity, and hysteretic energy absorption capacity. Thus, ψ can be expressed as

ψ (μ_{c}, T, β) = \frac{R (μ_{c}, T, β > 0)}{R (μ_{c}, T, β = 0)} ⩽ 1.0

(18)

Derivation of the closed-form expression of ψ is difficult and beyond the scope of this investigation. For simplicity, the investigation focuses on evaluating ψ based on the RHA result database. Nonlinear regression analyses were performed using Table Curve 2D. Based on trend of the RHA results and testing of numerous mathematical equations, the following empirical expression was adopted

ψ = a + b \frac{\ln (T)}{T}

(19)

where a and b are the coefficients providing best-fit predictions and influenced by µ_c and β.

Table 3 presents the a and b values obtained from the regression analyses. Figure 10 compares the predictions from equation (19) with the data obtained from RHA. As shown, equation (19) successfully captures the nonlinear trend of ψ over the considered range of other parameters. It is observed from Figure 10 that the deamplification factor ψ decreases as period of the oscillator increases but to a much lower degree when period of the oscillator exceeds 0.5 s. Moreover, Figure 10 shows that ψ tends to be lower in a system with higher µ_c and β values.

Table 3.

Parameters for determination of ψ.

µ _c	β = 0.1		β = 0.2		β = 0.3
µ _c	a	b	a	b	a	b
2	0.920855	−0.00273	0.871313	−0.00387	0.837325	−0.00431
4	0.872994	−0.00405	0.798492	−0.0059	0.732627	−0.00834
6	0.843115	−0.00501	0.743794	−0.00851	0.675083	−0.01005

Figure 10.

Deamplification factor for consideration of cumulative in determination of R.

As discussed in section “R factor without consideration of cumulative damage,” most of the formulae proposed by past investigations provide reasonable estimates of the R factor when β is equal to zero. The deamplification factor, ψ, can be conveniently used to modify the predictions of these formulae to take into account the effect of cumulative damage (i.e. estimating R for the cases with β larger than zero). Note that the coefficients listed in Table 3 were developed for specific µ_c and β values. For a given system having µ_c and β different from the considered values but within the considered ranges, a and b may be obtained through linear interpolation of the parameter values recommended in Table 3. It is recognized that extrapolation may not be proper to determine a and b for a system with µ_c and/or β outside the ranges considered in this research. Moreover, ψ should not exceed 1.0.

Illustrative numerical examples

Two numerical examples were developed to demonstrate adequacy of the work from this investigation. The first example shows how the deamplification factor, ψ, can be incorporated into a formula proposed by past research and improves estimates of the response modification factors. Specifically, in this example, for the considered period, µ_c, and β values, the response modification factors were first predicted by the formula proposed by Hidalgo and Arias (1990) (i.e. the model with best goodness-of-fit identified in section “R factor without consideration of cumulative damage”); then these predictions were adjusted by ψ. The response modification factors predicted before and after adjustment were compared with those obtained from RHA. The R_RMS values (see equation (17)) were computed to compare the result differences. Detailed R_RMS comparisons are presented in Figure 11. As shown, R_RMS values are significantly reduced after ψ is applied, indicating that implementation of ψ to the existing formula improves estimates of the response modification factors.

Figure 11.

Comparison of R_RMS.

The second example demonstrates the adequacy of equations (15) and (19) for an EPP oscillator with parameters not explicitly considered in this research. The considered system was assumed to be subjected to the ATC/FEMA far-field ground motions and have µ_c and β equal to 3 and 0.15, respectively. Note that such a combination of µ_c and β is representative of reinforced concrete structures with moderate ductility (Ghosh et al., 2011). Equation (15) was used to estimate the damage contribution of the cumulative damage caused by hysteretic energy absorption in the system at its damage limit state. Figure 12 compares the prediction from equation (15) with the RHA results. As shown, equation (15) provides a reasonable estimate of the cumulative damage due to hysteretic energy absorption.

Figure 12.

Damage due to hysteretic energy absorption in an example system.

In addition, the formula proposed by Hidalgo and Arias (1990) together with ψ predicted by equation (19) was used to determine the response modification factors. Note that the parameters, a and b, for computation of ψ (see equation (19)) were determined through interpolation of the data listed in Table 3. Figure 13 compares the median response modification factors obtained from RHA with those predicted from the method developed in this investigation. As shown, improved estimates of the R factors can be successfully achieved.

Figure 13.

Response modification factors for an example system.

Further considerations

The expressions and methods proposed in this research are for prediction of central tendency of the response modification factors of EPP oscillators. For a specific system, the response modification factor can exhibit great variations from one ground motion to another. This means that the base shear required for seismic design of a system to achieve controlled damage can have important variations from one ground motion to another. Thus, in some cases, using the response modification factors determined based on this research may not result in the desired level of safety. In those cases, it may be of help to use the response modification factors associated with higher level of probability of exceedance. To this end, the equations proposed in this research may remain the same but the parameters need to be re-evaluated based on the RHA result database and acceptable probability of exceedance. Note that the response modification factors based on this research (which adopts the central tendency as the quantity of interest) have a probability of exceedance of around 50%.

In addition, as shown in this research, cumulative damage plays an important role in determination of the response modification factors. Recent research (Hou and Qu, 2015) has demonstrated that ground motion duration has a significant impact on hysteretic energy absorbed and thus cumulative damage in a system. Although the ATC/FEMA far-field ground motions used in this investigation have a wide range of durations, ground motion duration was not explicitly considered as an individual parameter in this research. Future research opportunities exist to develop ground-motion-duration-dependent response modification factors.

Furthermore, the response modification factors determined according to this research are based on statistical studies of oscillators with assumed characteristics. Extrapolation of the results to multiple-degree-of-freedom structures requires (1) the knowledge about damage distribution within a structure, and (2) abundant laboratory testing data for better estimates of the key parameters, µ_c and β.

Conclusion

This article investigates the influence of cumulative damage on seismic response modification factors. To take into account the damages, respectively, caused by ductility and hysteretic energy absorption, the Park–Ang damage index was considered. A large number of iterative RHA were conducted using the EPP oscillators with parameters selected to represent conventional civil structural systems. The as-recorded ATC/FEMA far-field ground motions were used as inputs for RHA. Based on the analysis results, the following conclusions can be drawn from this study:

When an EPP oscillator system reaching its damage limit state, about 10%–50% of the total damage is due to hysteretic energy absorption. With this level of contribution, it is improper to ignore the cumulative damage in determination of response modification factors. In addition, as a result of the cumulative damage, the maximum tolerable ductility demand in a system reduces to about 50%–90% of its ductility capacity determined under the monotonic loading condition, depending upon other characteristics of the system.

When ignoring the effect of cumulative damage, the formulae proposed by past investigations provide reasonable estimates of the response modification factors obtained from the numerical simulations of this research. Among all the formulae revisited in this research, the one proposed by Hidalgo and Arias (1990) best captures the trend of the RHA results.

When cumulative damage is considered, the response modification factors become lower (and much lower in some cases depending upon ductility capacity and hysteretic energy absorption capacity of the system). None of the previously proposed formulae is adequate to capture the effect of cumulative damage in prediction of the response modification factors.

The deamplification factor, ψ, defined as the ratio of the R factor with consideration of cumulative damage to that without consideration of this effect, can be used to produce improved estimates of the response modification factors. The deamplification factor is dependent upon ductility capacity, hysteretic energy dissipation capacity, and period of an oscillator system.

The two numerical examples demonstrate that the expressions obtained from nonlinear regression analyses in this research are adequate to quantify the cumulative damage due to hysteretic energy absorption in a system reaching its damage limit state.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research reported in this paper was supported by the Open Research Fund of Sichuan Province Key Laboratory of Seismic Engineering and Technology, Southwest Jiaotong University, China, under Award No. SKZ2012001 and the General Program of National Natural Science Foundation of China under Award No. 51578324. Moreover, the first author (H.H.) was supported by the Fundamental Research Funds of Shandong University (No. 2014JC041) and the second author (B.Q.) was supported by the Tom and Lucia Chou Fund. The authors wish to acknowledge the sponsors. However, any opinions, findings, conclusions, and recommendations presented in this paper are those of the authors and do not necessarily reflect the views of the sponsors.

References

Araya

Saragoni

(1980) Capacity of the strong ground-motion to cause structural damage. In: Proceedings of the 7th world conference on earthquake engineering, Istanbul, Turkey.

ASCE/SEI 7-10 (2010) Minimum design loads for buildings and other structures.

Bertero

Anderson

Krawinkler

. (1991) Design guidelines for ductility and drift limits. Report no. UCB/EERC-91/15 EERC. Berkeley, CA: University of California, Berkeley.

Bertero

Mahin

Herrera

(1978) Aseismic design implications of near-fault San Fernando earthquake records. Earthquake Engineering & Structural Dynamics 6: 31–42.

Chopra

(2001) Dynamics of Structures: Theory and Application to Earthquake Engineering. Upper Saddle River, NJ: Prentice-Hall.

Cornell

(1997) Does duration really matter? FHWA/NCEER Workshop on the National Representation of Seismic Ground Motion for New and Existing Highway Facilities (NCEER project 106-F-541 and ATC-18-1), Burlingame CA, 29–30 May, pp. 125–133. Washington, DC: National Academies of Sciences, Engineering, and Medicine.

Federal Emergency Management Agency (FEMA) (2003) NEHRP Recommended Provisions and Commentary for Seismic Regulations for New Buildings and Other Structures (FEMA 450). Available at: http://www.nehrp.gov/pdf/fema450provisions.pdf

Federal Emergency Management Agency (FEMA) (2009) Quantification of building seismic performance factors (FEMA P695). Available at: https://www.fema.gov/media-library-data/20130726-1716-25045-9655/fema_p695.pdf

Ghosh

Datta

Katakdhond

(2011) Estimation of the Park-Ang damage index for planar multi-storey frames using equivalent single-degree systems. Engineering Structures 33: 2509–2524.

10.

Hidalgo

Arias

(1990) New Chilean code for earthquake resistant design of buildings. In: Proceedings of the 4th US national conference on earthquake engineering, Palm Springs, CA, 20–24 May.

11.

Hou

(2015) Duration effect of spectrally matched ground motions on seismic demands of elastic perfectly plastic SDOFS. Engineering Structures 90: 48–60.

12.

Kunnath

Reinhorn

Lobo

(1992) IDARC version 3.0: A program for the inelastic damage analysis of reinforced concrete structures. Report no. NCEER-92-0022, 31 August. Buffalo, NY: National Center for Earthquake Engineering Research.

13.

Lai

Biggs

(1980) Inelastic response spectra for aseismic building design. ASCE Journal of the Structural Division 106: 295–310.

14.

Miranda

(1993) Site-dependent strength reduction factors. ASCE Journal of Structural Engineering 119: 3503–3519.

15.

Miranda

Bertero

(1994) Evaluation of strength reduction factors for earthquake-resistant design. Earthquake Spectra 10: 357–379.

16.

Nassar

Krawinkler

(1991) Seismic demands for SDOF and MDOF systems. Report no. 95, June. Stanford, CA: The John A. Blume Earthquake Engineering Center, Stanford University.

17.

Newmark

Hall

(1973) Seismic design criteria for nuclear facilities. Report no. 46. Washington, DC: Building Practices for Disaster Mitigation, National Bureau of Standards, U.S. Department of Commerce.

18.

Park

Ang

AHS

(1985) Mechanistic seismic damage model for reinforced concrete. ASCE Journal of Structural Engineering 114: 722–739.

19.

Porush

(1988) Current changes and future trends in U.S. seismic code provisions. In: Proceedings of the 9th world conference on earthquake engineering, Tokyo-Kyoto, Japan, vol. V, 2–9 August, pp. 568–574.

20.

Rahnama

Manuel

(1996) The effect of strong motion duration on seismic demands. In: Proceedings of the 11th world conference on earthquake engineering, Acapulco, 23–28 June.

21.

Riddell

Newmark

(1979) Statistical Analysis of the Response of Nonlinear Systems Subjected to Earthquakes, (Structural research series no 468). Champaign, IL: University of Illinois at Urbana–Champaign.

22.

Riddell

Hidalgo

Cruz

(1989) Response modification factors for earthquake resistant design of short period structures. Earthquake Spectra 5: 571–590.

23.

Samanta

Megawati

Pan

(2012) Duration-dependent inelastic response spectra and effect of ground motion duration. In: Proceedings of the 15th world conference on earthquake engineering, Lisbon, 24–28 September.

24.

Saragoni

(1985) Azimuthal effects on earthquake destructiveness. In: Proceedings of the transactions of the 8th international conference on structural mechanics in reactor technology, vol. M, Brussels, 19–23 August, pp. 303–308.

25.

Van de Lindt

Goh

(2004) Earthquake duration effect on structural reliability. ASCE Journal of Structural Engineering 130: 821–826.

26.

Vidic

Fajfar

Fischinger

(1992) A procedure for determining consistent inelastic design spectra. Workshop on nonlinear seismic analysis of RC structures, Bled, 13–16 July.