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STRENGTH REDUCTION FACTORS FOR DUCTILE STRUCTURESWITH PASSIVE ENERGY DISSIPATING DEVICESDANNY ARROYO-ESPINOZA a & AMADOR TERAN-GILMORE aa Universidad Autonoma Metropolitana, Depariamento de Materiales , Av. San Pablo 180, Col.Reynosa Tamaulipas, Mexico, 02200, D.F, MexicoPublished online: 03 Jun 2008.

To cite this article: DANNY ARROYO-ESPINOZA & AMADOR TERAN-GILMORE (2003) STRENGTH REDUCTION FACTORS FORDUCTILE STRUCTURES WITH PASSIVE ENERGY DISSIPATING DEVICES, Journal of Earthquake Engineering, 7:2, 297-325, DOI:10.1080/13632460309350450

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Journal of Earthquake Engineering, Val. 7, No. 2 (2003) '297-325 @ Imperial College Press @ Imperial College Press w w w . ~ c ~ r e s s . c o . u k

STRENGTH REDUCTION FACTORS FOR DUCTILE STRUCTURES WITH PASSIVE ENERGY DISSIPATING DEVICES

DANNY ARROYO-ESPINOZA and AMADOR TERAN-GILbIORE Universidad Autonoma Metropolitona, Departamento de Materiales,

Au. San Pablo 180, Col. Reynosa Tamaulipas, MeTico 02200, D.F., Mexico

Received 7 July 2001 Revised 1 May 2002

Accepted 29 July 2002

In recent years, current seismic codes started contemplating the design of structures with passive energy dissipating devices. One important issue for the rational seismic design of these devices and the structure that contains them is the formulation of numerical methods to estimate their design seismic forces. Ftom the study of the dynamic response of single-degree-of-freedom systems subjected Lo accelerograms recorded in Mexico du- ring the last two decades, expressions to estimate the strength reduction factor that should be used to reduce the elastic design strength spectra for 5 percent damping, to establish the design seismic forces for structures having different combinations of plastic and viscous energy dissipating capacities, are formulated.

Keywords: Strength reduction factor; ductile structure; energy dissipation; supplemental damping.

1. Introduction

Since a long time ago, it has been accepted that the lateral strength that should be provided to an earthquake-resistant structure to enable it to survive intense ground motion diminishes as its plastic deformation capacity increases. According to this, the seismic design of standard structures contemplate the possibility of providing them with a significant plastic deformation capacity, in such a way that the design lateral strength can be kept within.a range of values that allow the design of the structure from an economical point of view.

One of the consequences of allowing the structure to undergo significant plastic behavior during a ground motion is the appearance of severe structural damage. Al- though the level of structural damage is controlled in such way as to avoid collapse, significant plastic demands can lead to highly undesirable situations, such as pro- hibitive structural rehabilitation costs and, in some cases, to excessive deformations that may result in excessive nonstructural damage.

Recent seismic events, such as Northridge 1994 ,and Kobe 1995, have empha- sized some of the problems associated to the use of traditional structural systems for earthquake resistance durhg severe ground motions [Northridge Reconnaissance

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Team, 19961, [Mitchell et al., 19961, [Bruneau and Yoshmura, 1996). As a con- sequence, the use of innovative structural systems has captured the attention of researchers and practitioners in recent decades. Within this context, an attrac- tive option is the addition of supplemental damping to structures through the addition of passive energy dissipating devices. The good seismic performance of these systems has been observed during experimental and analytical research [Aiken et al., 19931, [Hanson, 19931, [Miyasaki and Mitsusuka, 19921, [Whittaker and Con- stantinou, 19991, as well as in the field during recent seismic events [Northridge Reconnaissance Team, 19961.

Although several researchers have suggested that the use of supplemental d a m p ing should keep the structure nearly elastic during intense ground motion, this may

- -.not~be~economically~feasible,~p&i~~1arly for the seismic ?habilitation orlpgrad- ing of an existing structure. Furthermore, FEMA 274 [I9971 states in its section C2.10.7; " I n order to dasszpate substantial energy, dissapatzon dewices m w t typzcallg undergo significant deformation . . . Therefore, these systems are most effective i n structures that are relatively fleSbZe and have some inelastic defonnation capaczty." Thus, the addition of damping, through the addition of passive energy dissipating devices, may sometimes be focused on controlling the level of structural damage, not on eliminating altogether plastic behavior in the structure. In cases like this, the structure should be designed taking into consideration its plastic and vlscous energy dissipating capacities. One important issue for the seismic design of such structures is the determination of their design seismic forces. This implies, within the context of current seismic codes, the need to formulate strength reduction factors to reduce the code-defined elastic strength spectra for 5% damping according to the expected plastic behavior of the structure and the damping provided by the passive e n e r a dissipating devices.

An important issue for the seismic design of systems that provide supplemental damping to structures is the numerica! characterisation of their energy dissipating capacity. In this respect, FEMA 273 Guidelines for the Seismic Rehabdatation of Buildzngs [FEMA 273, 19991 and its associated commentaries [FEMA 274, 19973 state: "Passive energy devices add damping (and sometimes s t i f iess) to the build- ing's structure . . .". FEMA 273 considers that a wide variety of passive energy dis- sipating systems can be characterised, for design purposes, through an equivalent damping coefficient. As discussed in detail by Chopra [1995], this coefficient is usu- ally estimated by equating the energy dissipated in a vibration cycle of the actual structure and that dissipat& by an equivalent viscous system. Within this con-

_ - text, F E U 273 &ssifies - - passive -- energy dissipatihg devices into three categories: - - - -

displacement dependent (e.g. friction and metallic yielding devices), velocity depen- dent (e.g. viscoelastic solid, viscoelastic fluid and viscous fluid devices, and other shape memory alloys, friction-spring self centering assemblies, and fluid restoring force damping devices); and provides guidelines to estimate an equivalent damping coefficient for structures having either displacement or velocity dependent energy dissipating devices.

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Strength Reduction Factors 299

In this paper, the passive energy dissipation capacity will be characterised through an equivdent viscous damping coefficient (or equivalent damping coeffi- cient) in such way that the results and conclusions summarised herein apply to those systems that, like velocity and displacement dependent devices, can be characterised in this manner.

2. Preliminary Consideration

2.1. Definition

Consider a ductile structure with period of vibration (T), that is capable of under- going, during the design ground motion, a maximum plastic deformation demand characterised through its lateral displacement ductility ratio (p). In this paper, p is defined as the ratio between the maximum lateral displacement demand in the structure and its lateral displacement at yield. Furthermore, consider tbis struc- ture has passive energy dissipating devices, whose level of energy dissipation can be characterised through an equivalent damping coefficient (6). Within this context, the strength reduction factor, RpE, for such system will be expressed as:

RpC(llo,50, To) = c,b = l , S = 0.05,To)/~y(po,<oJo)

where c, denotes seismic coefficient, defined as the design base shear of the structure normalised by its total weight; p,, to, To are the specific values of p, and T associated to the system for which RpF is being determined, p of 1 implies elastic behaviour, and c, ( p = 1, < = 0.05,T0) is the seismic coefficient corresponding to the minimum strength that. would ,keep the structure within its elastic range of behaviour during the design ground motion provided it has en equivalent damping coefficient of 5%. Although the dehition in Eq. (2.1) applies to systems having any hysteretic behaviour, this paper concantrates on the dynamic response of systems exhibiting elastc-perfectly-plastic behaviour.

2.2. Previous studies

The dynamic response of an elasto-perfectly-plastic single-degree-of-freedom (SDOF) system subjected to ground motion strongly depends on the interaction between its mechanical characteristics and the dynamic characteristics of that mo- tion. Within this context, the frequency content and duration are characteristics of the seismic excitation that need to be carefully established to obtain reasonable estimates of the seismic demands and overall dynamic response of the structure. Because the RPE of a system strongly depends on its dynamic response; while the dynamic characteristics and duration of a ground motion, on the type of soil in which it is generated; the magnitude and overall dependency of RA on j . ~ and T, is significantly influenced by the type of soil in which the design ground motion is generated [Nassar and Krawinkler, 19911, [Miranda, 19931, [Ordaz and Perez, 19981,

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[Miranda, 2000], [Arroyo, 20011. Through the work of these and other researchers, the following general trends have been observed:

Independently of the type of soil, RPE tends to one as T approaches zero, and to p for very large T. For motions generated in firm soil, the R,€ corresponding to SDOF systems hav- ing # of 0.05 and developing a maximum ductility demand of p during the ground motion, increases rapidly when starting from zero, the value of T is increased. After peaking with a value slightly larger than p at a T somewhat larger than the predominant period of the excitation (T,); RA exhibits a slight decrease with a further increitse in T until it reaches a value of p for large T. For motions generated in soft soil, the RPc corresponding to SDOF systems hav-

~- -- ing < of 0.05 and developing a maximum ductility demand of pi increases rapidly when starting from zero, the value of T is increased; until it peaks with a value considerably larger than p at T equal to T,. After it has peaked, Rut decreases at a high rate 4 t h a further increase in T k t i l it reaches a value of for large

- -. ...- - -- -- --.- - - . - . -~

T. The over& dependency of RA on T can be better expressed, for regression purposes, if T is normalised by T,. R,< is fairly independent of the duration of the ground motion, and of other important characteristics, such as its intensity and epicentral distance.

0 A variation of the frequency content of the seismic excitation may or may not have an important effect on the value of RpE. Particularly, the values of RA cor- responding to firm soil are not affected in an important manner by the frequency content of the motion (i.e. the type of firm soil); while those corresponding to soft soil can be affected sigdicantly by a variation in the frequency content of the motion.

2.3. Conceptual scope

In the preliminary stages of the research project reported herein, the viability of using passive energy dissipating devices to control the response of ductile systems wa studied. From the response of SDOF systems with different combinations of p and < and subjected to motions with different frequency content and duration, Arroyo [2001] notes that although the effectiveness of supplemental damping in reducing the maximurn response of an earthquake-resisting structure decreases as p increases, an increase in < results in an important increase in RrSi even for large values of p (e.g. 4). Fkom these observations, it was concluded that a combination of -- - -- - - - - - -- - - -- - .- - - - plastic and viscous energy dissipating mechanisms represents a viable alternative, from a structural point of view, for earthquake resistance. Thus, it was decided to formulate expressions to estimate RPc, defined according to Eq. (2.1), for different combinations of p and < and different type of ground motions. Based on what was learned during the previous studies, the following goals were established:

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Strength Reduction Factors 301

Formulate two expressions for R,<, one for firm soil and another one for soft sdil. To accomplish this, two sets of 152 ground motions were established; one for each type of soil. For each ground motion within the sets, the RPF for SDOF systems having different combinations of T , p and < was established. Then, the average (i.e. the mean) value of R,€, corresponding to all the motions within one set, was established for each combination of T, p and J. Then, regression analyses, based on the least square method, were carried out to fit curves to the mean values of RpE, in such way as to minirnise, within reasonable practical constraints, the quadratic error of the fit. The expression for firm soil was to be formulated as a function of T, p and <, and would not take into account explicitly the frequency content, duration, intensity and epicentral distance of the ground motions. The expression for soft soil was to be formulated as .a function of T/T,, p and J, and would not take into account explicitly the frequency content, duration, intensity and epicentral distance of the ground motions. Nevertheless, given the observations made by Arroyo 12001j regarding the influence of the frequency content on the values of R,€ in soft soil, some extra studies were carried out to study this influence, in such way as to discuss the applicability of the expression formulated in this paper to any type of excitation generated in soft soil. Each one of the sets of motions was to be established in such way that they included several low intensity motions and a few high intensity motions. These motions were grouped with no consideration to their intensity, duration and epi- central distance. Within this context, the only consideration done was to avoid including impulsive type of motions in the sets.

2.4. Numerical scope

Two separate expressions to estimate R,€ are presented in this paper, one for firm soil and the other one for soft soil. These expressions were derived from the response of single-degreeof-freedom (SDOF) systems with elasto-perfectly-plastic behaviour. Values of p ranging from 1 to 4 and of ranging from 0.02 to 0.30 were considered. Regarding the range of < selected, it should be mentioned that in one hand, the lower bound of 0.02 represents the equivalent damping coefficient for welded steel, prestressed concrete and well-reinforced concrete working at serviceability stress levels [Cliopra, 19951. In the other hand, while values of E ranging from 0.10 to 0.15 have been achieved for passive energy dissipating systems in Mexico City; there is analytic and experimental evidence that suggest that values of < as high as 0.30 can be achieved in structures having a wide range of T [Miyasaki and Mitsusuka, 19921, (FVhittaker and Constantinou, 19991. Finally, it should be noted that the linear analysis procedures specified by FEMA 273 (to be discussed in detail later) for the analysis of structures having passive energy dissipating devices, are limited to systems with a J of 0.30.

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For f k n soil, T values ranging from 0.1 to 4.0 s were considered, as well as 152 accelerograms recorded in different f h n soil sites located throughout the pacific coast of Mexico-and the hill zone of MeAco City. These motions were filtered to eliminate low frequencies and avoid base line problems; this process was carried out using the DEGTRA 2000 program [Ordaz and Montoya, 2000). This set of accelerograms included motions with peak ground accelerations ranging from 0.0059 to 0.459, and was characterised by several low intensity motions and a few high intensity motions.

For soft soil, TITg values ranging from 0.1 to 3.2 were considered, as well as 152 accelerograms recorded in Merent sites of the lake zone of Mexico City. The motions were filtered to avoid baseline problems using the Hodder filter [Hodder, 19831. Tg was defined as the value of T in which the elastic input energy spectra for < - o f - O ~ 0 5 - p e a k s ~ T h e - s e t - i n c l u d e d - m o t i o - t h - p e a k ~ ~ e l ~ t i o n s r a n g i n g from 0.01g to 0.179, and with Tg ranging fiom 0.8 to 4.7 s. Again, the set was characterised by several low intensity motions and a few high intensity motions.

Detailed information of all accelerograms included in the study and their filtering - - - - -is-presented in Arroyo-(20011. Figure 1 shows the mean-nomalised~elastic2stiength

spectra for < of 0.05 and for the two sets of motions. The normalised strength spectra, for each ground motion, was obtained by nomalising the strength spectra by ii, ,,/g, where ii,, is the peak ground acceleration and g the acceleration of gravity. While for firm soil the spectrum peaks at a T close to 0.3 s; the spectrum corresponding to soft soil peaks at TIT, equal to 1.0. Note that the average spectra for soft soil exhibits a second peak at TIT, cIose to 0.5, which is a consequence of the fact that several motions within the set exhibited more than one peak in their strength spectra.

FEMA 273 classifks a soil profile, according to the average shear wave velocity in its upper 30 meters (u,), into the categories summarised in Table 1. Miranda

0 1 2 3 T(s) 4 0 - 1 2 3 TITg

(a) Firm soil (b) Soft soil

Fig. 1. Average elastic strength spectra, < = 0.05.

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Strength Reduction Factors 303

Table 1. Types of soil according to FEMA 273.

Type of soil va ( m / 4

A - Hard Rock Greater than 1.500 B - Rock 75&1300 C - Very dense soil and soft rock 370-750 D - Stiff soil 181t3iO E - Soft day with more than 10 feet of depth Less than 180 f - Special soils Not specified

[2000] has observed that the influence of the soil type in the inelastic displacement ratio (closely related to the strength reduction factor) is very small, and can be neglected for design purposes, for sites with an average shear wave higher than 180 m/s. With this in mind, firm soil refers in this paper to any of the four soil profiles in Table 1 having an average shear velocity larger than 180 m/s (e.g. soil types A, B, C and D). Regarding the motions recorded in soft soil? it should be mentioned that the soft soils in Mexico City have shear wave velocities typically in the range of 65 to 75 m/s, and have very high water contents. This type of soil is not contemplated in classifications such as that summarised in Table 1, in such way that a special characterisation needs to be carried out to capture their high level of seismic energy around their dominant period of excitation, and the narrow band of frequencies around this period.

3. Reduction Factors for Firm Soil

Once the set of ground motions for firm soil was established, a regression analysis was carried out to formulate an expression that, as a function of the variables involved in Eq. (2.1), yields reasonable estimates of the mean value of RPE . The following family of curves was used:

The basic form of Eq. (3.1) is defined by the expression To/(@ + .TQ), which tends to zero as the value of T approaches zero, and tends to one for large values of T. The peak value of R,( depends of C$ and 9, while the value of B defines the rate of decrease of RPC once it has peaked. Through a regression analyses based on the least square method to reduce the maximum quadratic errors, the following expressions for q!~ and d were found:

(3. lb)

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Fig. 2. Comparison between real and estimated [Eq. (3.1)) mean values of R,.c, firm soil.

Figure 2 compares the real mean values of R,€ (continuous lines) with those obtained using-Eq. (3.1) (discontinuous Lines) for p of 1, 1.5, 2, 3 and 4. As shown, there is a good coincidence between both sets of values (although not shown, the coincidence is just as good for other values of c, e.g. 0.02).

The practical application of Eq. (3.1) may be deemed too complicate, partic- ularly because of the estimation of 8. Practical expressions to estimate R,€ must consider a balance between required accuracy and simplicity. To illustrate this, consider a simplification of Eq. (3.1) according to the following:

Figure 3 compares the real mean d u e s of R,< (continuous lines) with those obtained using Eq. (3.2) (discontinuous lines). It can be seen that the prediction of RpE, particularly for the combination of large < and small p, although reasonable,

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Strrngth Reduction Factors 305

Fig. 3. Comparison between real and estimated [Eq. (3.2)] mean values of RPe, firm soil.

is not as good as that obtained with Eq. (3.1). A quantitative comparison between the accuracy of Eqs. (3.1) and (3.2) can be established through the values of the logarithmic deviation (q,) and logarithmic error ( E L * ) associated to the estimates of R,< obtained from both equations. These measures of accuracy are defined as:

N M

€1. = 2.. N M . . k.2) t=l j=l

where N is the number of accelerograrns analysed; M , the number of periods con- sidered; Rij , the real value of RPs; and R:j, an estimgte of RPe obtained from either Eq. (3.1) or Eq. (3.2). Note that EI , has a positive value when, on average, the real value of R,€ is underestimated; while it has a negative value when, on average, the real value of R,€ is overestimated. While an undekstimation of R,< leads to

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306 D. Anvyo-Espinoza @ A. Temn-Gilmo~e

an overestimation of the required design strength, and thus to conservative seismic design, a significant overestimation of RPS can lead to unsafe design.

Tables 2 to 5 summarise the values of 01, and €1, for several combinations of p and <. As shown, the values of u1, corresponding to Eq. (3.1) tend to be srnder than those computed from Eq. (3.2), difference that becomes more noticeable as the values of p and { decrease. A similar trend can be observed for the values of E,,, . On average, Eq. (3.2) tends to overestimate the value of RPE. This overestimation is characterised, according to Table 5, by values of ~1, ranging £rom -0.010 to -0.085. As a reference, a value of -0.085 implies R,€ is overestimated by a factor of e0.085 = 1.089. Equation (3.1) also tends to overestimate, on average, the value of R,<. According to the values surnmarised in Table 3, the largest overestimation wilI be by a factor of e0.063 = 1.065, while the largest underestimation, by a factor 0f-e=0.035--- - - 0~966.

The value of €1, is a measure of bias (an EI, equal to zero implies an estimation is unbiased). As may be concluded from the values summarised in Tables 3 and 5, Eq. (3.1) yields, on average, slightly less biased estimates of R,€ than Eq. (3.2). Both Eqs.-(3:l) and (3.2) yield reasonable estimates of R&; and thus, c a f b e used for reliable seismic design. When making a decision of the format that will be

Table 2. Logarithmic deviation, Eq. (3.1).

5 p = l p = 1 . 5 p = 2 p = 3 p = 4

- - - -

Table 3. Logarithmic error, Eq. (3.1).

Table 4. Logarithmic deviation, Eq. (3.2). - -- - -- - -

c p = 1 p = 1 . 5 p = 2 p = 3 p = 4

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Strength Reduction Factors 307

Table 5. Logarithmic error, Eq. (3.2)

adopted for an equation to estimate RPE within the context of a building code, it is important not only to consider the accuracy of such expression, but to take into consideration the opinion of the practical engineers that will apply it in practice.

4. Reduction Factors for Soft Soil

Once the set of ground motions for soft soil was established, a regression analysis was carried out to formulate an expression that, .as a function of the variables involved in Eq. (2.1), yields reasonable estimates of the mean value of RPc. The following family of curves was used;

The basic form of Eq. (4.1) is defined by the expression (T/Tg)/(b+ /T/Tg - I/) , which was formulated so that it would be similar to the basic expression formulated for firm soil. The basic form for soft soil tends to zero as the value of T approaches zero, and leads to much larger values of RPc than those estimated by the basic form for firm soil for values of T close to T,. The peak value of RPc depends of q5 and 0, while the value of 8 defines the rate of decrease of RPc once it has peaked. Through a regression analyses based on the least square method to reduce the maximum quadratic errors, the following expressions for 6 and 0 were found:

(4. lc)

Figure 4 compares the real mean values of R,( (continuous lines) with those obtained using Eq. (4.1) (discontinuous lines) for ,u of 1, 1.5, 2, 3 and 4. As shown, there is a good coincidence among both sets of values (although not shown, the coincidence is just as good for other values of c, e.g. 0.02). It should be mentioned that the real mean values of RPC corresponding to the first peak in the strength elastic spectrum Isee Fig. l(b)], are larger than those obtained with Eq. (4.1); nevertheless, trying to account for this difference would lead to a more complex expression than Eq. (4.1). As discussed before for firm soil, there is a need to

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Fig. 4. Comparison between r e d and estimated [Eq. (4.l)j mean R,<, soft soil.

balance the complexity and required accuracy for the formulation of an expression to estimate RA.

5. Dispersion and Other Levels of Response

Earthquake-resistant design is usually not carried out using mean seismic demands. The reliability involved in seismic codes requires a dsign associated to other levels of response, which may be associated to the mean plus a given number (n) of standard deviations (a) demand. Within this context, it is not only of interest to formulate, through the mean value of R,g, a relation between the mean elastic strength spectra for 5% damping and the mean strength spectra corresponding to different c o m b i n a t i o n s - ~ f - p - ~ d - < t b - u t - t o - e ~ t a b h l t i ~ between the mean + nu strength spectra for 5% damping, and the mean+un strength spectra corresponding to different combinations of p and 5. Thus, it becomes important to evaluate the scatter of RPr around its mean value; and to discuss the impact of this dispersion on the earthquakeresistant design of ductile system having passive energy dissipating devices.

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Strength Reduction Factors 309

One way to account for the scatter of RPE is to use a lower percentile of R,< to reduce the elastic strength spectra for 5% damping (e.g. mean - no). Another way in which this may be accomplished is to use the mean value of RPC to relate to other levels of response according to the following:

- - mean + n&(p = 1,[ = 0.05, T = To)] (5.1) mean+no[c,(p = p a , < = eo ,T = To)]

where n is the number of standard deviations considered. The latter approach, which has been studied in a different context by Teran-Gilrnore [1996], suggests that the mean value of a ratio involving the means of two seismic demands, may be used, if n is small, to obtain a reasonable estimate of the ratio involving an upper percentile of those two seismic demands (the same percentile for both). Although

(a) 5% damping

(c) 208 damping

I I I I

0 1 2 3 T(s) 4

@) 10% damping

COV 0.8 $4.30

Fig. 5. COV of strength reduction factors for motions recorded in firm soil.

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310 D. Arroyo- Espinoza E4 A . Temn-Gilmote

the approach implied by Eg. (5.1) will be discussed in detail in this paper, imple- mentation details for the practical use of both approaches will be discussed. The dispersion of R,< will be quantified through the coefficient of variation (COV), defined as the standard deviation of the response quantity of interest, normalised by its corresponding mean. Figure 5 summarises the COV values obtained for R,< corresponding to the motions recorded in firm soil.

Figure 5 shows that for all values of damping considered, the COV tends to increase as the level of plastic deformation increases; nevertheless, while an increase in damping tends to have little effect in the value of COV for large p, this effect tends to increase significantly as the value of p decreases. As consequence of this, the value of COV for p of I increases significantly with an increase in 6 ; while for p of 4, and increase in < has practically no impact on the value of COV. Note that-this-can-be-e~~lained-in-~art-b&],

Mean - a ~~ Fig. 6. Different percentiles of RPC for motions recorded in firm soil.

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Strength Reduction Factors 31 1

which results in a COV of zero for elastic systems having a < of 5%, Also, it can be noticed that with the exception of very short periods, the value of COV is fairly independent, for a given value of p. of T. Figure 5 also shows the COV spectra exhibits a noticeable but narrow peak at a T around 0.1 s. Independently of the value of I, the highest COV found in this study (if the peak is not considered), is around 0.35 for a p of 4. The values of COV sumrnarised in Fig. 5(a) are similar to those obtained, for firm soil and 5 percent damping, by other resedrchers for ratios involving the means of two seismic demands [Miranda, 20001.

As a consequence of the tendencies shown in Fig. 5 for the COV, which is practically independent of T, other percentiles of RPS show similar tendencies, with respect to the values of T, p and <, than those exhibited by its mean. Figure 6, which shows the mean - u , mean, and m e a n + a of R,<: illustrates this fact for different combinations of p and J.

As Fig. 6 suggests, the ratio between the m e a n + a and m e a n values of Rd is around 1.5 for all combinations of p and { included in the figure, and is fairly independent of T. This suggests that an expression to obtain estimates of other percentiles of RPc would not be much more complicate than those surnrnarised in Eqs. (3.1) and (3.2). The observations made above can be considered as an extension of observations made previously by other researchers. For instance, Whittaker et al. [I9981 have suggested, .within a slightly different context than the one established in this paper, that the ratio between the m e a n + a and m e a n values of a ratio involving two seismic demands can be characterised by a value of 1.5. Based on this, it can be said that expressions similar to those included in Eqs. (3.1) and (3.2) could be formulated to estimate any percentile of RPc, provided a multiplicative factor is added to account for the variability of RPc.

Figure 7 illustrates the results obtained, for different combinations of p and e l by applying the approach summarised in Eq. (5.1), to obtain m e a n + u strength spectra for firm soil. The results shown in this figure were obtained from a 10 motion subset of the original set of 152 motions used to establish Eqs. (3.1) and (3.2). These motions were recorded during different seismic events, in such way that they have very different peak ground acceleration (PGA). To provide them with comparable intensities, in such way that they can represent, as a set, a design ground motion, all 10 motions were scaled to have a PGA equal to that exhibited by the NS component of El Centro 1940 (0.359).

The observations and conclusions derived from the discussion to follow are in- dependent of the value of PGA selected to scale the ten ground motions. Within this context, the PGA corresponding to El Centro NS 1940 was selected just a s a reference value, taking into consideration that this motion is a well known motion recorded in firm soil. Three curves are shown in Figs. 7(a) and 7(c) according to the following: the thick continuous line corresponds to the real mean +a strength spectra for the given combination of p and t; the thin continuous line corresponds to an estimate of the mean +a strength spectra, for the same p and 4, obtained by

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(a) Strength spectra, p = 4 and 4 = 0.05 (b) Strength ratio, p = 4 and ( = 0.05 - - . . - - -

(c) Strength spectra, p = 1 and F = 0.20 (d) Strength ratio, p = 1 and E = 0.20

Fig. 7. Real and estimated mean f a strength spectra for motions recorded in firm soil.

plugging into Eq. (5.1) the real mean value of R,€; and the discontinuous thin line corresponds to an estimate of the mean + a strength spectra, for the same p and J, obtained by plugggg into Eq. (5.1) the mean value of RpE estimated by using Eq. (3.1).

Figures 7(b) and i ( d ) show the real strength spectra shown in Figs. 7(a) and 7(c), respectively, normalised by their corresponding estimates. As before, the thin continuous line corresponds to the estimate obtajned by using the real mean'value of RrE, while the thin discontinuous h e corresponds to the estimate obtained by using the mean value of Rp< estimated from Eq113.1). AS suggested by the_results - -. --

shown, the use of the approach summarked in Eq. (5.1) yields a reasonable estimate of the mean +a strength spectra for the different combinations of p and < shown in Fig. 7. Note that the use of the mean - a value of R,< to relate the mean + a elastic strength spectra for 5% damping and the mean + a strength spectra for a given combination of p and J, would result in very conservative estimates of the

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Strength Reduction Fncto7-s 313

(a) 5% damping

COV 0.6 1 I

COV 0.6 <a. 10

(b) lO%damping

COV 0.6

0 1 2 3 Trrg

(d) 30%damping

Fig. 8. COV of strength reduction factors for motions recorded in soft soil.

strength demands; and that the results shown in Fig. 7 have been obtained from a small subset of the original set of 152 motions.

Figure 8 summarises the COV values obtained for RPE corresponding to soft soil. As shown, for all values of damping considered, the COV tends to increase as the level of plastic deformation increases; nevertheless, as was the case for firm soil, an increase in damping tends to have little effect in the value of COV for large p, while it has a significant effect for small values of p. As suggested before, this is explained in part by the definition of RpE- Also, it can be seen that in the majority of the cases under consideration, the value of COV is fairly independent, for a given value of p, of T. The exceptions to this observation can be found for SDOF systems with small <, particularly for those developing a small p.(whose COV tends to increase in an important manner as the value of T approaches that of T,); and for SDOF systems with large < and develop& a small p (whose COV tends-to decrease in -

an important manner as the value of T approaches that of T,). Note that with the

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314 D. Anuyo-Espinoza €4 A. Teran-GiLm

Fig. 9. Different percentiles of RPc for motions recorded in firm soil.

exceptions of the cases just mentioned, the value of COV for the motions on soft soil have a similar magnitude than that exhibited by the COV corresponding to firm soil.

As consequence of the fact that the COV for soft soil tends to be fairly indepen- dent of T for a given value of p, other percentiles of Rpt show similar tendencies, with respect to the values of T, p and <, than those exhibited by its mean. Figure 9, which shows the mean - a, mean, and mean + o of Rpt, illustrates this fact for different combinations of p and 5. Thus, as discussed before for firm soil, it can be

s~d-that-an-expression-similar-to-that-included-in-Eq(471-)-can-be-fomulated-to--- estimate any percentile of Rd. Nevertheless, the use of a constant multiplicative factor, would lead, depending on the percentile under consideration, to slight over and underestimation of the value of Rpc for SDOF systems with T close to T,.

Figure 10 illustrates the results obtained, for different combinations of p and (, by applying the approach summarised in Eq. (5.1), to obtain mean -+ a strength

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Strength Reduction Factors 315

(a) Strength spectra, p = 4 and 6 = 0.05

C,, 0.700 + I

(b) Strength ratio, p = 4 and e = 0.05

0 1 2 3T/T, 0 i 2 3 TIT,

(c) Strength spectra, p = 1 and < = 0.20 (d) Strength ratio, p = 1 and < = 0.20

Fig. 10. Real and estimated mean + u strength spectra for motions recorded in soft soil.

spectra for soft soil. The results shown in this figure were obtained from a 12 motion subset, having a T, of 2 s, of the original set of 152 motions used to establish Eq. (4.1). These motions were recorded during different seismic events, in such way that they have very different PGA. To provide them with comparable intensities, all 12 motions were scaled to have a PGA equal to that exhibited by the EW component of SCT 1985 (0.179). The observations and conclusions derived from the discussion to follow are independent of the value of PGA selected to scale the ground motions. Within this context, the PGA corresponding to SCT EW 1985 was selected just as a reference value, taking into consideration that this motion is a well known motion recorded in soft soil. The curves shown follow the same convention established for Fig. 7. As suggested by Fig. 10, the use of the approach surnmarised in Eq. (5.1) yields a reasonable estimate of the mean + u strength spectra for different combinations of p and <. Note that the results obtained by

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using the mean value of R,< estimated from Eq. (4.1) does not yield adequate results for T/T, close to 0.5, which can be explained by the fact that this equation has not been formulated to capture the peak exhibited in the R,: spectra for TITg equal to 0.5. Note that again, the use of the mean - a value of R,( to relate the mean + a elastic strength spectra for 5% damping and the mean + a strength spectra for a given combination of p and E would result in conservative estimates of the strength demands; and that the results shown in Fig. 10 have been obtained from a small subset of the original set of 152 motions.

6. Other Issues

6.1. Boundary conditions

E q u ~ a t i o ~ r i s ~ ( 3 ~ ~ ) ~ d - ( 4 ~ l ~ ) ~ ~ ~ u ~ d ~ ~ a ~ i ' s ~ * ~ h ~ ~ f o l l ~ i ~ b ~ u ~ d a r y condi%iZn~:-R;i~ -

should tend to one as T approaches zero; while it should tend to p for large T. Both equations satisfy the first condition; while none of them satisfy the second one. Regarding this, it should be noted that none of the equations were formulated to-estimate-the value-~f-R;~-for very.large T. - ~ - - ~- -- - -- - - - -

A third condition that should be satisfied by both equations, this one derived fiom the definition presented in Eq. (2.1), is that R,< should be equal to one for p equal to 1 and 5 equal to 0.05. For these values of p and 5, Eqs. (3.1) and (4.1) oscillate between 0.96 and 1.00, and between 1.00 and 1.02, respectively, which is considered satisfactory.

6.2. Interaction between p and < An accurate estimate of R,< for a structure that simultaneously dissipates plastic and viscous energy requires the explicit consideration of the interaction between

(a) Firm soil (b) Soft soil

Fig. 11. Comparison between estimates of R,( obtained by neglecting and accounting for the interaction between p and E , < = 0.30.

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Strength Reduction Factors 317

p and <. Figure 11 compares, for a < of 0.30, the estimates of R,[ obtained by neglecting this interaction (discontinuous lines) and by taking into account such interaction (continuous lines). To neglect the interaction means that two separate and independent reduction factors are formulated, one to account for p and the other one for <, and the value of R,€ is obtained as the product of these two factors; while to account for the interaction means to estimate R,< according to Eq. (2.1). As shown in Fig. 11, neglecting the interaction between p and 5 results in an overestimation of RpE, particularly for systems built on soft soil and having T/T, close to one. On one hand, the overestimation for firm soil is relatively small and may be neglected without introducing significant error during seismic design; while in the other hand, the overestimation cannot be neglected on soft soil and TITg close to one.

6.3. EBect of frequency content on soft soil

For soft soil, the value of RPe exhibits important variations with respect to the frequency content of the ground motion. Figure 12 compares the mean R,€ for 5 of 0.05, p of 1.5, 2, 3 and 4, and two subsets of the initial set of accelerograms for soft soil: the first made up of motions with T, of 2 s; and the second, of motions with T, of 4 s. It should be noted that the bandwidth of the second set around T, is smaller than that of the first set. As shown, although the maximum value of RPE is similar for both subsets, R,< decreases at a considerably higher rate for T/Tg larger than one for the second subset. Thus, the conformation of a set of soft soil motions to obtain expressions for IIPc needs to be done carefully. As discussed before, the conformation of the set of motions recorded on soft soils and the expression proposed to estimate the mean RPt for this set need to consider the balance between the accuracy required during the design process and the ease of use in a practical context.

0 1 2 3 Tfls 0 1 2 3 T f f ,

Fig. 12. RA for soft soil motions with different frequency content, of 0.05.

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6.4; Comparison with other formulations

While Figs. 13 and 14 compare, for several combinations of p and <: the real mean values of R&, estimates of these values obtained by using Eqs. (3.1) and (4.1), and estimates obtained using other well known proposals [Wu and Hanson, 19891, (Miranda, 19931, [Ordaz and Perez, 19981, [FEMA, 19991; Tables 6 to 9 present the logarithmic errors associated to these estimates. Note that the comparison that is established should be interpreted carefully, because the equations formulated by

Wu --3t Eq. (3.1) - - - -Avg I F E M A

Fig. 13. Comparison of different estimates of R,( for firm soil.

- - I - Avg Eq. (4.1) - - -X- - - Miranda Ordaz

Fig. 14. Comparison of different estimates of RPc for soft soil.

- Table 6- -Logarithmic errom for-firm soil, .Wu.and Hanson, . - - - - -

E ~ = 1 p = 1 . 5 p = 2 p = 3 p = 4

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Strength Reduction Factors 319

Table 7. Logarithmic errors for firm soil, FEMA 273.

t p = l

Table 8. Logarithmic errors for soft soil, Eq. (4.1).

Table 9. Logarithmic errors for soft soil, Miranda and Ordaz ( E = 0.05).

p = 1 p = 1 . 5 p = 2 p = 3 p = 4

Miranda 0.000 0.045 0.029 -0.024 -0.070 Ordaz 0.000 0.004 -0.039 -0.118 -0.176

the different researchers have been obtained using different sets of ground motions. As shown in Fig. 13, Eq. (3.1) yields similar or slightly better estimates of RPE

than other proposals for firm soil, while providing a sounder and more general basis for estimating RpE for system having different combinatiom of p and <. Tables 2, 6 and 7 summarise the logarithmic errors associated to the estimates obtained from Eq. (3.1) and the other proposals for firm soil. From the value of these errors, it can be concluded that Eq. (3.1) yields reliable estimates of RPE.

Similar conclusions can be derived for Eq. (4.1) from Fig. 14 and Tables 8 and 9. It should be mentioned that Ordaz and Perez [1998] proposal yields better estimates of Rpe for TIT, around 0.5 [location of a second peak, as shown in Fig. l(b)], and that RPe in Fig. 14(b) was defined according to Ordaz and Perez [1998], and not according to Eq. (2.1), to make possible the comparison shown in that figure.

7. Issues in Earthquake Resistant Design

According to FEMA 273, the analysis of structures having displacement or/and velocity dependent energy dissipation devices, can, be carried out by characterising, through an equivalent damping coefficient, the energy dissipatiou capacity of these

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320 D. Arroyo-Espinoza €4 A. Temn-Cifmom

devices. The damping coefficient used for this purpose can be estimated as:

where tea is the damping coefficient for the structure with the energy dissipating devices; 5, the damping coefficient in the structure without these devices (usually set equal to 0.05); W j , the work done by device j in one complete cycle corresponding to the expected floor displacements & (the summation extends over all devices j); and Wk, the maximum strain energy in the structure, determined as follows:

where Fi is the inertia force at floor level i (the summation extends over all floor levels).

FEMA 273 provides several options to carry out the structural analysis of a structure with energy dissipating devices. Among them, the Linear Procedures - - _ _ _ _ _ . _._ -- - - limited to structures whose effective damping does not exceed 30% of critical in the fundamental mode, and that remain (exclusive of the energy dissipating devices) linearly elastic for the selected level of earthquake hazard. Two Linear Procedures are included in FEMA 273:

Linear Static Procedure (LSP). The LSP is based on a static analysis of the building under equivalent static lateral forces, and is limited to regular structures with height lower than 30 m; Linear Dynamic Procedure (LDP). Based on response spectra or time history dynamic analysis that considers more than 90% of the reactive mass of the struc- ture, the LDP is applicable to a wider range of buildings. In this case, values of teff should be estimated and applied to each mode considered in the analysis.

Within the context of the linear procedures described above, the design seismic forces should be derived horn a strength spectrum that should be reduced, as a function of the value of &, according to what is specsed in Sec. 2.6.1.5 of FEMA 273.

In the next paragraphs, some of the observations and conclusions derived in Secs. 3 to 6 of this paper will be used from a critical perspective, to discuss the linear analysis procedures allowed by FEMA 273. Two important issues that need to be addressed are: f ist , the strength reduction factors formulated by FEMA 273, as a function of tee, are independent of the type of soil at the construction site; and -- .- - second, ChaptT9-6fFEMA-274 states thmhFKEnergy disszpation devices will also reduce force in the structure-provided the structure is responding elastically- but would not be ezpected to reduce force in structures that are responding beyond yield".

Regarding these two issues, the results obtained in this paper suggest that: first, the strength reduction due to &a varies significantly according to the type of soil (of

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Strength Reduction Factors 321

PLAN VIEW

(a) Geometry (b) Design spectra

Fig. 15. Geometry of 10-storey building and design spectra.

particular importance within this context is the correct numerical characterisation of motions generated in soft soil); and second, supplemental damping is effective in reducing the strength demands even in systems developing significant plastic demands.

Witbin this context, a third and very important issue raised by FEMA 273 Guidelines is the limitation of use of the LSP and LDP to structures that remain (exclusive of the energy dissipating devices) linearly elastic. To address this issue, a 10-storey building [Fig. 15(a)] has been designed according to the expressions derived in this paper, and its dynamic response studied. The structural elements of the building were dimensioned in such way that its T is equal to 2 s, and it was assumed that the building has energy dissipating devices in such way that teff is equal to 0.20 (FEMA 274 offers design examples of frames with passive energy dissipating devices that may be of interest to the reader). As a design condition, it was assumed that the building is capable of undergoing a p of 2 during the design pound motion. The design elastic strength spectrum for < of 0.05 (associated to the mean plus one standard deviation demands), shown in Fig. 15(b), was obtained from 12 ground motions recorded in soft soil and having Tg of 2 s (same subset used to establish Fig. 10). All these motions were scaled to have a peak ground acceleration of 0.179. Figure 15(b) also shows the strength design spectrum for p of 2 and < of 0.20, obtained by dividing the elastic spectrum corresponding to < of 0.05 by Eq. (4.1). From the design strength spectrum for p of 2 and ( of 0.20, a seismic coefficient, G, of 0.14 is obtained for a T of 2.0 s (i.e. for T/Tg equal to one).

Two versions of the building were designed. one derived &om a static analysis and having a base shear (Vb) of 0.14 W (where W is the total weight of the building), and one derived from a dynamic analysis and having a Vb of 0.112 W (80% of the static base shear). A nonlinear model was established for each version for analysis with DRAIN 2DX [Prakash et al., 19931. For each version, the value of Va discussed above is the ultimate base shear the building is able to develop during a pushover

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322 D. Arroyo-Espinoza E4 A. Temn- Gilmme

analyses with a relative lateral load distribution through height obtained from a dynamic modal analysis. In the examples discussed herein, all the uncertainties in- volved in determining the seismic supplies in the buildings have been eliminated; that is, the buildings have been designed for given values of T, ( and c,, and the DRAIN 2DX models of the buildings have been established in such way that the values of these mechanical characteristics are equal to their design values. Within a practical context, this is unlikely; that is, given all the uncertainties and com- plexities involved, the fundamental period, energy dissipation capacity and lateral strength of the structure will likely differ from the values used during its design. Bertero and Bertero [2000] note that within the context of seismic design, the uncertainties associated to the definition of the response spectra are considerably

-- larger-than-th~e-associated-to-the-determinati~n-of-the.mechanical~characteri~cs of the earthquake-resistant structures; in such way that the overall reliability of the design is not affected significantly by the uncertainty involved in determining the mechanical characteristics of the building. - - The dynamic response of each version of the-building, when subjected to each

one of the 12 ground motions used to establish Fig. 15(b), was estimated through a time-history analysis in which the first two modes of the building were assigned a of 0.20 (Rayleigh damping). Table 10 summarises the results of these analy- ses. The ductility demands in the table were estimated by dividing the maximum global lateral displacement demand during the ground motion by its correspond- ing global lateral displacement at yield. To establish the lateral displacement at yield for each building, a pushover analysis was carried out. Using the definition of equivalent SDOF system model established by Qi and Moehle (19911, it was found that the displacement of the seventh storey of the building has a direct correspon- dence to that of the equivalent SDOF system, in such way that the maximum global lateral displacement demands and the buildings lateral displacement at yield were estimated according to the value of the lateral displacement at this storey. According to this, the results of the pushover analysis were used to establish a global lateral displacement (at the seventh storey) vs. base shear curve, and the yield displacement of the building was defined as that one that delimits the elastic and plastic ranges of a bilinear idealisation of this curve. The maximum global lat- eral displacement demand for a given ground motion was defined as the maximum absolute lateral displacement demand in the seventh storey of the building for that motion.

- . - - - - - - - - - - . _ - -

Table 10. Global ductility demands in lbstorey building.

Displacement at Building h yield (cm) p(kean) p(rnean + cr)

--

Static Design 0.14 W 0.185 1.39 1.63 Dynamic Design 0.112 W 0.148 1.66 1.98

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Strength Reduction Factors 323

As may be concluded from the mean and mean plus one standard deviation values of the p demands sumrnarised in Table 10, the use of linear procedures of analysis in conjunction with Eq. (4.1) yields reliable earthquake-resistant design. Particularly, for the design derived from the dynamic method of analysis, a mean+a ductility demand of 2 is obtained, which is the condition for which the building has been designed. The design derived from the static method of analysis has a mean+a ductility demand of 1.6, which outlines the conservative nature of this method, particularly when the first mode of vibration dominates the dynamic response of the structure.

&om the response of the 10-storey building, it can be concluded that the LSP and LDP are viable alternatives for the analysis of structures that, in spite of the presence of passive energy dissipating devices, exhibit moderate plastic demands. That is, the LSP and LDP should not be limited to structures that remain iinearly elastic when subjected to the design ground motion. The large gap that exists in FEMA 273 between the linear and nonlinear analysis procedures for structures having energy dissipating devices, should be bridged by extending the application of linear procedures to structures that exhibit moderate plastic demands.

The energy demands associated to motions recorded in soft soils are considerably larger than those associated to firm soil motions [Uang and Bertero, 19881. Thus, energy dissipation is particularly attractive for structures built on soft soil and having a fundamental mode of vibration close to the dominant period of the soil. The behaviour of buildings built on soft soil is dominated by its fundamental period of vibration in a wide range of T [Teran-Gilmore, 20021, in such way that even the response of buildings with 20 to 30 storeys is dominated by its first mode. Within this context, the LSP would be perfectly applicable to the analysis of these buildings in spite that their height may be well in excess of the limiting height of 30 m specified by FEMA 273.

Passive energy devices are an attractive and clean alternative for the rehabilita- tion of existing buildings. It should be noted that rehabilitation schemes are usually conceived to address and correct structural irregularities in the existing building. and to control plastic demands in its structural elements. Within this context, the analysis of rehabilitated structures can be carried out in many cases by using LSP and LDP (regular structures with moderate plastic demands), particularly for low and midrise buildings built on firm soil, and for a wider range of buildings built on soft soil.

Finally, it is important to emphasise that the use of an equivalent damping coef- ficient to characterise the energy dissipation capacity of some devices, particularly of displacement dependent devices, may lead in some cases to an oversimplification of the structural model used in the analysis. Thus, it is important, as FEMA 273 Guidelines require, that a separate analytical evaluation be performed to confirm a design meets each goal of the design objectives.

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8. Conclusions

In recent decades, various energy dissipating structural devices have been developed and even implemented in earthquake-resisting structures. An important issue for the seismic design of these type of structures is the formulation of strength reduction factors that allow for a reliable estimation of their design lateral seismic forces from an elastic design strength'spectra associated to a 5 of 0.05.

It is important and possible to formulate expressions to accurateIy estimate the value of RA for ductile structures having passive energy dissipating devices. These expressions should take into account the type of soil, the expected plastic behaviour of the structure (which may be characterised by p), the energy dissipating capacity provided by the devices (characterised by c), the period of the structure, and in the case of soft soils, the interaction between the dynamic characteristics of the struct<pre and those of the ground motion. Because the effectiveness of the supplemental damping decreases as p increases, it is very important to take into account, in the formulation of such expressions, the interaction between p and c.

A reliable seismic design of a structure requires the consideration of the scatter - - - - - . - - - - - - . - . - - - - - -. - . - . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

of R,,€ around its mean value. The use of a lower percentile of R,{ to reduce the design elastic strength spectra for 5% damping (associated to the mean + nu strength demands), to obtain the mean f nu strength design spectra for a given combination of p and 5 usually yields very consenrative results. A better approach, illustrated in this paper through the design of a lbstorey building, is the use of the mean value of R,< to establish the mean + nu strength demands from the mean + nu design elastic strength spectra.

The frequency content of a motion recorded in soft soil may have an important effect on the value of R,€, particularly for systems having a T slightly larger than T,. Thus, in some cases, the conformation of a set of motions to formulate an expression to estimate R,{ needs to consider carefully the frequency content of the motions recorded in soft soil.

Further research needs to be carried out regarding the use of energy dissipating devices in ductile structures, and the way in which their design strength demands can be established. Among other issues, the following need to be addressed: the formulation of expressions to estimate the mean value of R,€ for other types of soils; the characterisation of an energy dissipating structural system through a value of 5; validation of the two approaches described in this paper to establish the design strength spectra for a given level of response; and to,establish the technical and economical viability of using passive energy dissipating devices to control structural damage in new and existing ductile structures.

References

Aiken, I. D., Nims, D. K., Whittaker, A. S. and Kelly, J. M. I19931 "Testing of passive energy dissipation systems," Earthq. Spectrn 9(3), 335-370.

Arroyo, D. [2001] "Factores de reduccidn de resistencia para sistemas pasivos de disipaci6n de energia," Master thesis, Universidad Nacional Autbnoma de Mexico, Mkxico.

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Strength Reduction Factors 325

Bertero, R. D, and Bertero, V. V. [ZOO01 "Application of a comprehensive approach for the performance-based earthquake-resistant design of buildings," Proc. Twelveth World Conf. Earthq. Engrg., Paper 0847.

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