ELSEVIER PII: S0167-4730(96)00011-2

Structural Safety Vol. 18, No. 2/3, pp. 195-224, 1996 Copyright © 1996 Elsevier Science Ltd.

Printed in The Netherlands. All rights reserved 0167-4730/96 $15.00 -e .00

Dual-level designs of buildings under seismic loads

Y.K. W e n *, K.R. Col l ins l, S.W. Han 2, K.J. E l w o o d 3

Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, 1L 61801, USA

Abstract

Three methods of dual-level seismic design for buildings are proposed based on consideration of reliability against the serviceability and ultimate limit states. The first is a method of calibration of design parameters according to the two target reliability levels within the context of current code procedures. The method is demonstrated by calibration of seismic load factor and drift limit. The second is an extension of the design earthquake approach to allow consideration of two levels of design earthquake. The performance is assured by linear and nonlinear equivalent static (pushover) analyses. It produces designs which come closer to the design objectives of most current codes. The third is an alternative to the current design procedures. It uses uniform hazard response spectra and a pushover analysis-based equivalent single-degree-of-freedom system to explicitly account for variability and uncertainty in the seismic loads and inelastic response behavior of the structures. This avoids the use of the controversial response reduction factor. Uncertainties in the soil amplification and structural system modeling are accounted for by corresponding design factors similar in concept to the LRFD method. It ensures that the structure meets local (interstory) drift and global ductility requirements with specified probabilities. Copyright © 1996 Elsevier Science Ltd.

Keywords: Reliability; Seismic; Codes and standards; Calibration; Limit states; Dynamics; Nonlinearity; Pushover analysis; Response spectra; Simulation

I. Introduction

The building industry has suffered serious setbacks after the 1994 Northridge earthquake in the USA and the 1995 Great Hanshin earthquake in Japan. The earthquake resistant designs in these two countries are the most advanced and yet in Kobe, damages cost over a $100 billion and life loss reaches into the thousands. In Los Angeles the death toll would have been much higher had the earthquake struck during the day time and on a weekday. These two events bring to focus the serious questions of how the structural engineering profession treats the large uncertainty of seismic loads and the reliability of current building design procedures against such loads. Although the uncertainty of

* Corresponding author. Current address: University of Michigan, Ann Arbor, MI 48015, USA.

2 Current address: Hanyang University, Seoul, Korea. 3 Current address: Buckland and Taylor Ltd, North Vancouver, BC, Canada.

195

1 9 6 Y.K. Wen et al.

ua

b 0.1

0 . 0 1 ' ' '

0.5 1 1.5 2 2.5 Local (lnterstory) Drift ( % )

Fig. 1. Comparison of 50-year interstory drift exceedance probabilities: Sendai, 7-story RC moment frame (×); Los Angeles, 5-story steel SMRSF (O); Tokyo, 7-story RC moment frame (line); Taipei, 8-story steel EBF (+); Imperial Valley, 5-story steel SMRSF ([]).

seismic loadings has been well recognized by the profession, the incorporation of uncertainty in most building code procedures has been limited to the selection of design earthquake based on probability. For example, the design earthquake is defined in terms of a 10% exceedance probability in 50 years (or a return period of 475 years) in most US codes. This design earthquake is then used in conjunction with a series of factors reflecting the influence of structural period, site soil condition, structural inelastic behavior, importance of the structure etc. These factors are largely determined based on judgment and experience and often calibrated in such a way that the resultant designs do not deviate significantly from the acceptable practice at the time. Therefore, despite their simplicity and ease of use, a significant shortcoming of the current design procedures is that the reliability of the final design is undefined and difficult to quantify.

Recently, several studies [1-3] have been carried out with the objective of evaluation of the reliability of buildings designed according to current code procedures under seismic loads. Buildings were designed according to the Uniform Building Code [4] in the USA at two representative locations in California, according to the 1992 (PREcast Seismic Structral System) PRESSS Guidelines in Japan [5] in Tokyo and Sendai, and according to the Taiwan Code in Taipei. Ground motions of future earthquakes were generated according to seismicity of the regions and stochastic ground motion models which depend on the source, path and site characteristics. Realistic structural models are used and repeated nonlinear time-history responses were calculated. The performance of the buildings were evaluated in terms of the probability of various interstory drift levels being exceeded in 50 years. A summary of these studies has been given by Wen [6]. Fig. 1 shows the performance of the foregoing buildings. Fig. 2 shows the seismicity of the sites in terms of the 50-year exceedance probability of peak ground acceleration. It is well understood that performance of structures of different construction materials and different framing systems cannot be measured by a single response measure such as interstory drift. Nevertheless, these performance curves give valuable information of the reliability of current code procedures in countries where seismic loads have been an important design considera- tion. The results show large variation in the performance which can be attributed to differences in site seismicity, design philosophy, construction material and coefficient values in code provisions such as the drift limit and response reduction factor to count for inelastic response capacity. It is obvious that

Y.K. Wen et al. 1 9 7

~= 0.1

o

0.01 0 0.2 0.4 0.6 0.8

Peak Ground Acceleration ( g )

Fig. 2. Comparison of seismic risks (50-year PGA exceedance probabilities) at five sites: Taipei (+); Sendai (×); Los Angeles (~); Tokyo (line); Imperial Valley ([]).

current code procedures cannot be expected to produce designs which satisfy probabilistic perfor- mance goals or criteria based on risk/benefit trade-off.

To achieve such risk explicit design goals, the consideration of uncertainty in the design process needs to be extended beyond the selection of the design earthquake. Since structures generally become nonlinear and inelastic under severe earthquakes, accurate description and modeling of structural systems in the nonlinear range are required. In such a risk (or reliability)-based design, target performance curves similar to those in Fig. 1 may be used as the design criteria. They can be chosen depending on the function and importance classification of the structure. The objective of the design is then to ensure such performance curves are satisfied at least at some key points corresponding to, for example, serviceability and ultimate limits. Alternatively, performance objec- tives such as operational, immediate occupancy, life safety and collapse prevention [7] can be used for this purpose. The target reliability associated with each performance level, however, needs to be specified.

Design methods based on this philosophy have been recently developed. In the following, three dual-level design methods with consideration of the serviceability and ultimate limit states are proposed. The first is a method of calibration of design parameters according to target reliability levels within the context of current code procedures. The second is an extension of the design earthquake approach to allow consideration of two levels of design earthquake, i.e. one for the serviceability limit state and one for the ultimate limit state. The performance is assured by linear and nonlinear equivalent static (pushover) analyses. Comparison of structural performances based on this approach and conventiona! methods is made using the extensive ground motion data of the 1994 Northridge earthquake. The third represents a departure from the current design procedures. It uses uniform hazard response spectra and a pushover analysis-based equivalent single-degree-of-freedom system to explicitly account for variability and uncertainty in the seismic loads and inelastic response behavior of the structures; thus it avoids the use of the controversial response reduction factor. Uncertainties in the soil amplification and structural system modeling are accounted for by corre- sponding design factors similar in concept to the Load and Resistance Factor Design (LRFD) method. It ensures that the structure meets local (interstory) drift and global ductility requirements with specified probabilities.

198 Y.K. W e n et al.

2. Bi-level calibration of design parameters in current code procedures

2.1. Background

Within the context of the current code format, one can adjust the factors and coefficients in the provisions in such a way that the resultant design will have desirable reliabilities against specified limit states. It is commonly referred to as "code calibration". In current code procedures, the design earthquake is determined based on probability which obviously has an impact on the reliability of the design. In addition, the load factors which account for the overall uncertainty in the loading, the importance factor which accounts for the different levels of performance required of the buildings and the drift limits will also affect the reliability of the design. The design earthquake and these factors and limits are, therefore, the logical targets for calibration. They are also interconnected as far as the overall reliability of the structure is concerned such that selection of one of these design values without considering the others may lead to inconsistent reliability. The only rational method of determining the design values is calibration according to explicit target reliabilities against specified limit states. Factors based on consideration of structural dynamics, soil condition, ductility capacity and so on need not be subjected to calibration.

Within the current code format, one can select key reliability-related factors as design variables represented by a vector X and proceed with the code calibration process by minimizing the difference between the target reliability and that of the design as follows [8]:

O= E E wq[ pi;(x) - pf ]2/Pj, (1) i j

in which i refers to type of structure and j refers to limit state under consideration; o),2 is the weight assigned to each case; Pij(x) is the probability of ith limit state of structure type j designed using the design variable vector X = x; and Pj* is the target probability of the jth limit state. One advantage of the Eq. (1) over alternative formulations (e.g. in terms of index /3 or log P) is that the penalty term is proportional to the failure probability. It allows one to assign the weights according to the consequences of limit state; thus it is especially suitable for calibration when multiple limit states of different target probabilities and consequences are considered such as serviceability and ultimate limit in the seismic design. In checking the safety of the design, it is most convenient to express the limit state function in terms of the load effect and structural resistance variables; or in other words the reliability problem is formulated in the load effect space. In formulating the design criteria, however, it is more convenient to directly specify the criteria in terms of load factors and other design limits. This is particularly true in code calibration where the conventional design formats are followed. For example, the LRFD format used in ANSI /ASCE 7-88 is a well-known example, though only member limit states were considered. In this study, the design variables X consist of the load factors and drift limits. The above minimization assures that the resulting load factors and design variables lead to designs which satisfy the prescribed target reliabilities for all limit states and types of structures under consideration as much as possible.

2.2. Design of steel moment frame buildings according to NEHRP

To illustrate the procedure, examples on calibration of design variables for steel frame buildings are given in the following. Details can be found in Han and Wen [9]. Since the earthquake load factor

Y.K. Wen et al. 199

Table 1 Structural design parameters

Design parameters Frame 1 Frame 2 Frame 3 Frame 4 Frame 5 Frame 6

No. of stories 12 6 3 4 2 5 Story height (ft) 12 (15) 14 (17) 12 (15) 14 (17) 14 (17) 12 (15) No. of bays (NS) 4 3 3 4 3 4 No. of bays (EW) 5 5 5 5 5 5 Span length (ft) 25 25 30 30 30 25 Dead load (psf) 100 95 100 95 95 100 Live load (psf) 50 45 45 50 45 50

Note: First story height is shown in parenthesis.

and interstory drift limit are the most dominant factors, the effort is concentrated on calibration of these two factors, i.e. X in Eq. (1); otherwise, the 1992 National Earthquake Hazard Reduction Program (NEHRP) procedures [10] are followed. Six Special Moment Resisting Steel Frame (SMRSF) structures of 2 - 6 and 12 stories are considered. A Latin hyper cube sampling method is used to select building configurations and dimensions which represent the steel building population. Table 1 shows the combinations of the structural design parameters of the six representative buildings. The buildings are located at a site in Los Angeles studied in Wen et al. [1]. The serviceability limit is chosen to be an interstory drift of 0.5% of story height and the ultimate limit is 1.5% of story height. The design earthquake has a return period of 475 years. In the design, the nominal dead load is 100 psf and nominal live load is 50 psf. The equivalent static base shear is calculated according to [10] using:

v= c,w,

1.2 A~S Cs= RT2/3 ,

in which A v period.

(2)

(3)

= 0.4 for Zone 4; R = 8 for SMRSF; S = 1.0 for dense and stiff soil; and T = structural

2.3. Calibration of design coefficients

Strictly speaking, solution of the minimization problem given in Eq. (1) requires nonlinear programming. The computation can become excessive since, in search for the minimum point, repeated designs and reliability evaluations are needed dependent on the number of iterations required. To alleviate this difficulty a Response Surface method with a central composite design is used to reduce the number of combinations of seismic load factor and drift limit to 9, hence limit the calculation required. The frames are assigned weights according to the total floor area of the building type and the ultimate limit state is assigned a weight much larger than that for the serviceability limit state to reflect the seriousness of the consequence of the limit states. In actual code calibration, the weights should be determined based on consensus among professionals experienced in assessing consequences of different limit states. The load combinations considered are those of sustained live and transient live, sustained live and earthquake, and all three time varying loads. The dead load is

2 0 0 Y.K. Wen et al.

assumed to be deterministic in these combinations for simplicity. The variation of live loads over time is simulated according to live load survey statistics [11]. The responses and conditional probabilities of exceeding limit states are evaluated by the time-history/simulation method [1]. As the occurrence behaviors of characteristic and non-characteristic earthquakes are different, the contributions to the overall limit state probability from these two sources are treated separately and then combined.

As expected, the combination of sustained live load with earthquake contributes the most. Although the combination of all three time varying loads leads to slightly higher limit probabilities, it also has a small joint occurrence rate because of the very brief durations of both the earthquake and the transient live load resulting in a very small contribution in the limit state probability. The limit state probability evaluation is most computationally intensive in this calibration effort since repeated response analyses are required for each frame for nine combinations of load factor and drift limits. An approximate method has been developed by which the response of a multi-degree-of-freedom (MDOF) system under seismic excitation can be represented by that of an Equivalent Nonlinear System multiplied by a global and a local response modification factor. The computation required can therefore be signifi- cantly reduced and becomes manageable. The equivalent system consists of two single-degree-of-free- dom (SDOF) inelastic structures representing the dynamic motion of the system, otherwise the

1 0 o

1 0 -1

10 - 2

I 0 - 3

10 - ~

o ~o a

, , , , l h , , , , , i , , , , | , i , , i l l

10 - 2 10 -1 1 0 o

Global Limit State Probability (Nonlinear MDOF System)

Z L~

:3 o'3

10 0

Z

10 -1

i~ lO-' . J

J

10 - 3

10 -~ 10 -2 10 -1 10 0 Local Limit State Probability (NMS)

Fig. 3. Comparison of global (building) and local (interstory) drift limit state probabilities of equivalent nonlinear system and nonlinear MDOF systems.

Y.K. W e n et al. 201

~tlW

"lhrget Probabilities :

/ 0.~6o (dtirostc) ~ ,, , I . . . . I . . . . I . . . . I . . . .

llwa,o

u

"lhrget Probabilities : . / ~ ,~, ,("I'~.~! °~. ,

Fig. 4. Scatter plot of objective function predicted by response surface method.

method is similar in concept to Enoue and Cornell [ 12]. Fig. 3 shows a comparison of the probabilities of exceedance of various drift thresholds for buildings of different heights using the approximate method with those based on the response analysis of the MDOF by the well known computer software DRAIN-2DX [13]. The data points are an amalgamation of 50-year exceedance probabilities of interstory and global drift limits ranging from 0.5 to 3% of building or story height for seven SMRSF steel buildings of 1, 2, 5, 9 and 12 stories. The method improves the accuracy of the equivalent system at slightly more computation effort. The RSM is found to work well and give close fits to the objective functions with an average error of 3%. Fig. 4 shows some sample comparisons of actual versus predicted values of the objective function.

2.4. Results and sensitivity analysis

The design earthquake load factor and the interstory drift limit (% of story height) obtained from the minimization of the objective function are shown in Table 2 for a few selected combinations of target limit state probabilities. The weight ratios assumed are also shown. Note that each combination corresponds to specifying two check points on the target probabilistic performance curve; therefore matching at these two points satisfies (at least approximately) the design performance requirements. The sensitivity of the design factors to the target limit state probability is also investigated. Fig. 5 shows the sensitivity to the target ultimate limit state probability and the relative weights assigned to the two limit states. It is found that depending on the relative weights, the design factors may be more

Table 2 Earthquake load factor (3'E) and allowable drift limit (A, % of story height) for various combinations of target 50-year limit state probabilities

Ps* 0.30 0.40 0.50 0.35 0.35 0.35 Pu* 0.06 0.06 0.06 0.04 0.05 0.06 YE 1.30 1.01 0.82 1.44 1.33 1.15 A 1.5 1.7 1.8 1.1 1.4 1.5

Ps* = Target probability for serviceability limit state; and Pu* = Target probability for ultimate limit state. Ratio of weights for serviceability versus ultimate limit state = 1:10. Ratio of weights for 2, 3, 4, 5, 6 and 12 story SMRSFs = 20:14:6:4:3:1.

202 Y.K. Wen et al.

1.50

1.40 ( ~ ~

1.3o C

1.20 ?E

1.10

1.00

0.90 X - - service:ultlmate= 1:15 0.60 i I i I i I ~ I ~ I

0.045 0,050 0.055 0.060 0.065 0.070 0.075 Pu*

Fig. 5. Sensitivity of seismic load factor to target ultimate limit state probability P.* serviceability to ultimate limit states.

for various weight ratios of

sensitive to one of the target limit state probabilities. The relative weights, therefore, need to be carefully considered, e.g. based on an assessment of the expected consequence of the limit states.

3. Design based on dual-level seismic forces

3.1. Background

A generally accepted earthquake resistant design philosophy is that buildings should resist small earthquakes with no damage, moderate earthquakes with limited and non-structural damage, and large earthquakes without collapse. In most building codes, however, it is only required that buildings be designed for one ultimate force level. Thus, in effect, the buildings are only designed for the third criteria. The exception is the "capacity design" concept adopted in the Japan [5] and New Zealand procedures [14]. In these procedures, the building is initially designed for serviceability under a moderate seismic force. Under a severe seismic force, the inelastic response behaviors are explicitly considered and additional design measures are incorporated to ensure satisfactory post-yielding structural performance such as prevention of a column yielding mechanism. The design earthquakes and the performance goals, however, are not clearly specified in terms of probability. Therefore, the risks implied are also unknown and undefined. A study [15] was recently carried out based on the concept of capacity design but with judicious choice of the two levels of design earthquake based on probability which may result in buildings that come closer to attaining the original design philosophy. Two seven-story reinforced concrete (RC) moment resisting frames, with three bays of 30 ft each in the short direction and seven bays of 20 ft each in the long direction, were designed for a site in Los Angeles. One frame was designed according to the 1992 US NEHRP Provision [10], while the other was designed according to the proposed dual-level design procedure. The latter generally follows the design philosophy of the "Ult imate Strength Design Guidelines for Reinforced Concrete Buildings", prepared by the Japan PRESSS Guidelines Drafting Working Group. Since both frames are located in the US, the ACI 318-89 Code [16] was used in both cases. Only major differences in design between these two procedures are mentioned. The method and results are summarized as follows. Design details can be found in [15].

Y.K. Wen et al. 203

3.2. Design of RC frames according to NEHRP

For the NEHRP frames, as is common in US design practice, only the perimeter frames were designed to resist lateral loads. The rest of the frames were designed to resist only gravity loads and the floor slab should distribute the seismic loads to the perimeter frames. The design earthquake is represented by an effective peak velocity-related acceleration (A v) for an earthquake with a 10% probability of exceedance in 50 years (or a return period of 475 years). The equivalent lateral force for base shear is determined according to Eqs. (2) and (3) in which a response modification factor R of 8 for moment frame is used. Dead and live loads and load combinations are determined according to BSSC 1991. The members were sized to ensure the interstory drift remained below the required limit of 0.15% of story height. Once the section dimensions were chosen to comply with the deflection requirements, a plane frame analysis program was used to compute the member forces and to choose the required reinforcing steel areas. The strong-column-weak-beam (SCWB) requirements of ACI 319-89 are also satisfied.

3.3. Design of RC frames using dual-level design earthquakes

In the PRESSS guidelines, two limit states are considered: serviceability and ultimate. For each limit state an equivalent lateral force is applied and certain performance objectives are satisfied. The design earthquake story shear at the ith story is calculated by:

Qi = ZRt A i C B W i , (4) in which

Z = the seismic zone factor; R t = the vibration characteristics factor taking into account types of soil; A; = vertical distribution of seismic story shear; and C B = standard base shear coefficient.

A structure should be serviceable after a moderate earthquake motion. The standard base shear coefficient for the serviceability limit state is 0.2. The shear is distributed along the structural height according to a nonlinear formula. No member is allowed to yield in this limit state and the story drift ratio must be less than 0.005. A structure should not collapse during a strong intensity earthquake motion. The standard base shear coefficient for the ultimate limit state is 0,3 for moment resisting frames in which reduction due to the structure's capacity for inelastic deformation is implicitly considered. At a story drift ratio of 0.01, which is called the "design limit deformation", the story shear at any story from a pushover analysis must be more than 0.9 times the design story shear for the ultimate state. At a story drift ratio of 0.02, which is called the "design proof deformation", the story shear must be more than the design story shear for the ultimate limit state. Furthermore, a SCWB design is confirmed by checking that all the yielding during the pushover analysis occurred in the beams and column bases. Regions other than the specified yield hinges must be designed not to yield. The design of these non-yielding regions must take into consideration factors which might increase the member design forces such as dynamic effects and bi-directional response. Therefore the basic philosophy of this design procedure is to avoid large plastic deformation, concentration of damage in

2 0 4 Y.K. Wen et al.

0.~6 I , , ~ l i , , , I I , , , I , , , , I , , , , I , ~ , , I , , , I . - - Uniform Hazard Elastic RS ( IS year R.P.)

0 . 2 0 Uniform Hazard Elastic RS ( 10 year R.P.) - - - - Design Spectrum (10 year R.P.)

0 . 1 5

0 . 0 5 m

0 . 0 0 . 5 1 .0 1 .5 2 . 0 2 . 5 3 . 0 3 . 5 P e r i o d (mec)

Fig. 6. Serviceability limit state design spectra for dual-level design.

limited locations, and brittle failure [17]. These performance checks avoid the use of general deflection amplification factor in the NEHRP provisions, and thus, allows the designer to satisfy the performance criteria by providing specialized detailing.

In contrast to the NEHRP design, in the dual-level approach each frame within the building was designed to resist the seismic forces. The frame designed for this study was a typical interior frame. The procedure generally follows the PRESSS guidelines. Since no specific probability levels are given for design earthquakes in the guidelines, the design earthquakes for the serviceability and ultimate limit states were chosen with the aid of the uniform hazard response spectra (UHRS) for the site with stiff soil (see Section 4.2) and on a trial-and-error basis. To remain consistent with the ultimate design level in US codes, a return period of 475 years was chosen for the ultimate design earthquake at a short period (0.3 sec) using the UHRS. The choice of the serviceability return period required more deliberation. The periods of 50, 15 and 10 years are considered. A 50-year return period was ruled out since keeping the structure elastic at such a high force level would require abnormally large member cross-sections. A 15-year return period would lead to a design base shear of 0.19 (see Fig. 6). The resultant design was subjected to the Sylmar record from the 1994 Northridge earthquake. The maximum interstory drift was found to be only one third of that of the NEHRP design with negligible permanent displacement. This response was deemed over-conservative for a ground motion considered to be nearly representative of the ultimate design earthquake. Thus a 10-year return period was chosen and the frame redesigned using a base shear coefficient of 0.1 for serviceability limit state as compared to 0.2 in the PRESSS guidelines for Zone 1 in Japan. For the ultimate limit state, a base shear reduction factor of 0.15 was chosen (Fig. 7) as compared to 0.3 implied in the PRESSS guidelines. The ultimate and serviceability design base shears for a typical interior frame were finally calculated as 307 and 223 kips, respectively.

The member design follows ACI 318-89 and the serviceability force level was used for the initial flexural design. The frame was modeled on DRAIN-2DX and a nonlinear pushover analysis was conducted. The resulting base shear-interstory drift curve for the first story is shown in Fig. 8. It is seen that the design complied with the serviceability performance criteria, i.e. remaining elastic at the serviceability design base shear of 223 kips and with a maximum interstory drift less than 0.5%. For the ultimate limit state performance criteria, since a nearly elastic-plastic pushover curve will result,

Y.K. Wen et al. 2 0 5

1.0

0 .9

0 .8

0.7

0.8

~ 0.5

~ 0.3

I

0 . !

11.0 0.11

, , , l , , , , l , J , , l , , , , l , , , , l , , , , l , , , -- 4s-- • ..... Uniform Hazard Elastic RS ( ~75 year R.P.) "

"lv~ " \ - - - Design Spectrum with k=l (4/5 year R.P.) I ~ ,,- . . . . . . Design Spectrum with k---0.15 (475 year R.P.)

i l l \\ }/ ',,,,

'/".,

0.5 1,0 1.5 2 ,0 ~ .5 3 ,0 3 .5 P e r i o d ( s e c )

Fig. 7. Ultimate limit state design spectra for dual-level design.

only one level of checking is necessary. It is seen that the final design of the dual-level flame complies with the ultimate performance criteria as well. Furthermore, the pushover analysis also demonstrates that the required SCWB collapse mechanism was achieved. The reinforcement and member dimensions for the two final designs are compared in Fig. 9.

3.4. Response and damage analysis

Time-history response analyses of the two flames under recorded ground motion excitation were carried out (using DRAIN-2DX) and compared. The moment curvature and moment axial force interaction relationship were developed for RC structural members based on proper material models

500 4 5 0

[~ . . . . s ~ s a b f l i t ~ l/m/t ntato pertorman~ a ~ c k ~ Puzhov~ Curvl

4OO

3 5 0

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150 ~ i

100 1' i , i

50 ' i

0 0 . 0 0 0 0 . 0 0 5 0 . 0 1 0 0 . 0 1 5 0 . 0 2 0 0 .025

F i r s t S t o r y D r i f t ( A / h )

F i g . 8 . L o a d - d e f l e c t i o n r e l a t i o n s h i p f o r d u a l - l e v e l f r a m e a n d p e r f o r m a n c e c h e c k f o r u l t i m a t e a n d s e r v i c e a b i l i t y l i m i t s t a t e s .

206

50"

20"

Beam

| 5"

Y.K. Wen et al.

30" 90"

30" I [

20"

Column Beam

28"

I Column

12#8

12#8

12#8

12#8

24#6

24#6

24#9

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12#9 [ 6#8 6#8 4#8

12#9 6#8 6#8 I 4#8 4#8 I

12#11 7#9 ~ _ ~

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7#9 7#9 4#10 4#1~ ~ ,

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12#11 i ~ ,

6#9 6#9 ~

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6#7 3#7

10#8 6#7 3#7

10#8 6#7 3#7

10#8 6#8 3#8

10#8 6#g 3#8

10#8

10#813#9

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Fig. 9. Comparison of final design of 7-story RC frames according

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12#8

3#~

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to (a) 1992 NEHRP and (b) dual-level procedures.

for the concrete and the reinforcement. Details and limitations of the adoption of this program for RC structures can be found in [15]. Since the main objective of this study is to compare two design approaches, the limitations are of secondary importance. The Sylmar record of the 1994 Northridge earthquake the E1 Centro record of the 1940 Imperial Valley earthquake, and the Castaic record of the 1971 San Fernando earthquake were chosen to approximately represent strong, moderate and small ground motions, respectively. Three response quantities are compared: global roof displacement, interstory drift and damage index. A variation of the damage index (DI) as a linear combination of damage due to excessive deformation and damage from repeated cyclic loading developed by Park et al. [18] is modified and used in this study. Since in the DRAIN-2DX model plastic rotations (i.e. deformations) only occur at the element ends, the damage indices are defined for each member end as follows:

0 p + / 3 E H

DI = - ~ E--~mo, , (5)

0p is the largest plastic rotation experienced by the hinge during the ground motion record; 0 u is the ultimate plastic rotation capacity of the member; /3 is a model parameter that reflects the effect of

Y.K. Wen et aL 207

cyclic loading on structural damage; E H is as the dissipated irrecoverable hysteretic energy, i.e. total area enclosed by the hysteretic loops. Thus, E H includes the effects of duration and low-cycle fatigue, and provides a good measure of the damage potential of earthquake ground motions [19]; E n .... is the irrecoverable hysteretic energy from a monotonic pushover analysis. Since damage at a local level does not provide useful information for the direct comparison of the overall damage experienced by two different frames, the local damage indices are weighted over the entire frame to obtain a frame damage index. Park et al. determined that Dlfram e _< 0.4 indicated repairable damage, Dlfram e > 0.4 indicated unrepairable damage, and Dlfram e > 1.0 indicated total collapse. To satisfy the current code requirements of life safety, the overall damage index for the frame must be less than 1.0.

3.5. Comparison of performance of dual-level and NEHRP designs

The Sylmar County Hospital record of 1994 Northridge earthquake is 16 km from the epicenter. This record is characterized by two very large acceleration pulses ( P G A = 0.91g). The Sylmar response spectrum and acceleration time history are shown in Fig. 10. The global drifts (i.e. the roof displacement as a fraction of the total height of the building) and local (interstory) drifts for both the NEHRP and dual-level designs subjected to the Sylmar record are shown in Fig. 11. The maximum global drift of the NEHRP design is 1.4 times that of the dual-level design, and both occur within the first displacement excursion, indicating the importance of the blast of the initial acceleration pulse. The lower stiffness of the NEHRP design is evident in the longer period of vibration. The longer period of vibration may also be partially explained by the larger amount of inelastic deformation experienced by the NEHRP design (see discussion of flexural hinges below). The maximum interstory drifts for the NEHRP design exceed that of the dual-level design. It is interesting to note that while the global drifts of both designs remained below 2% as a maximum limit recommended by Sozen [20], the interstory drifts for the second story exceeded 3% and 2% for the NEHRP and dual-level designs, respectively. This would appear to indicate a concentration of drifts in the lower stories.

Further understanding of the seismic performance may be gained by observing the distribution of flexural hinges throughout the two designs (Fig. 12). Flexural hinges are formed when the moment at the end of a member exceeds the specified yield moment. For several stories (2, 3, 4 and 6) the NEHRP frame hinges have formed across all four columns, indicating the formation of a strong- beam-weak-column collapse mechanism. Although collapse is not certain in dynamic oscillation; nevertheless, the formation of a possible collapse mechanism threatens life safety which is the primary goal of the NEHRP provisions - - to protect life safety during severe earthquake ground motion. Although hinges have formed in many of the center columns of the dual-level frame, no single story has hinges across all four columns (except at the base). Thus collapse is not imminent. Formation of the hinges in the beams, prior to the columns, allows for increased hysteretic energy dissipation and evenly distributes the interstory drifts over the height of the frame. The improved performance of the columns in the dual-level design may again be attributed to the performance check on the SCWB design.

The overall frame damage index for the NEHRP design is 0.98. This would suggest that the frame has experienced nearly total collapse, thus not satisfying the life safety requirement of the NEHRP provisions. It should be noted that the DRAIN-2DX model assumes unlimited ductility in each member, and thus, is not able to detect member failure due to exceedance of the ultimate rotation capacity. The damage index attempts to detect this form of failure, and then determines the effect in

208 Y.K. W e n et al,

O. f i

0 . 0

- 0 . 5

- 1 . 0

0 2 4 6 8 10 12 14 18 t8 20 22 24 26 28 80 = 0.5

0.3

~ o.o

~ - 0 . 3

o ~ -o.5

0 2 4 5 8 10 12 14 16 18 20 22 24 26 2B 30

lo, _

J L I , A L L , , . , . . , . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . o.oo VrdPv'rr,', " ~ . . . . . . .

- 0 . 2 5

- - 0 . 50 r ' ' ' l ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' l ' ' ' 1 ' ' ' l ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' l l l l l l

0 2 4 6 8 10 12 14 18 18 20 22 24 26 28 30

T ime (see)

8 . o . . . , . . . . , . . . . , . . . . , . . . . , . . . . t . . . . t . . ,

e.5 Sy lmar I I . . . . E1 Cenb'o

Z.O ] / ........... Cast .dr

1.0 q t

• 0.0 0.5 1.0 1,0 2.0 2.5 8.0 3.6 4.0

Period (sec)

Fig. 10. Time histories and response spectra of 1994 Sylmar record, 1940 E1 Centro record and 1971 Castaic record.

o.o2 I ~ - . . . . _ ! , . ~

-0.01

-0 .02 . . . . I , , , t , , , I , , . I , , , I , , , I , , , I , , , i , , , I , , , t , * , l , , , l ~ , , l ~ , , 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 "04 I - ' . . . . . . , . . . . . . . . . . . . ' . . . . . . . . . . , ' ' ' , . . . . . . . , ' ' ' , . . . . . . . , ' ' -I

~ ~ M . . . . mm~ ,P d e ~ 0 . 0 2 b- i,,| . . . . D u d Level d ,m~m -1

o . o o _ , : , , , . . . _ . . . . . . . . . . . . . .

° i " - -4 ~ -0 .02

0 Z 4 6 a t o 12 14 16 ia 2o Zz PA s o z a ao 'l~me (.ec)

Fig. 11. Comparison of global and local drifts for the NEHRP and dual-level designs subjected to the 1994 Sylmar record.

Y.K. Wen et al. 2 0 9

P

L r

L

t I •

Fig. 12. Comparison of distribution of flexural hinges of NEHRP and dual-level designs subjected to the 1994 Sylmar record.

the entire frame by weighting the individual member damage indices by the dissipated hysteretic energy. This explains why the time history of the global drift does not suggest the collapse of the frame (i.e. no large permanent displacements), while the overall frame damage index suggests that collapse is imminent.

The overall frame damage index for the dual-level design is 0.78. This would suggest that the frame has suffered significant structural damage and will be torn down, but has not collapsed and thus, has not threatened life safety [18]. Since the Sylmar record was chosen to represent severe earthquake ground motions, the above performance may be considered acceptable. It should be noted that the individual damage indices for the hinges at the base of the columns are greater than 1.0. These large values may be partially explained by the fact that an interior frame was used to represent the dual-level design, and thus, carries higher gravity loads than the exterior frame used to represent the NEHRP design. The higher gravity loads result in reduced ductility capacity, and in turn, higher damage indices. In short, the dual-level design performed much better than the NEHRP design for the Sylmar record.

The 1940 E1 Centro record has formed the basis for many aspects of current design codes. It was therefore chosen to represent moderate earthquake ground shaking. The acceleration time-history and response spectra are shown in Fig. 10. The global and local drifts for the NEHRP and dual-level designs are shown in Fig. 13. The maximum global drift for the NEHRP design is 2.0 times that of the dual-level design. The most obvious difference in the performance of the two designs is the permanent drift of approximately 0.4% that remains in the NEHRP design. It should be noted that the global drifts of both designs remain below 1%, and thus, should experience little or no structural damage. The maximum interstory drifts for the NEHRP design exceed those of the dual-level design. As in the global drifts, the NEHRP design experiences a permanent drift while the dual-level design remains plumb. Unlike the Sylmar record, the interstory drifts for the NEHRP design are nearly equal to the global drifts, suggesting that the drifts are well distributed over the height of the frame. Note that the maximum interstory drift exceeds 0.5% for both designs, thus indicating that non-structural damage may result from this ground motion. However, non-structural damage is generally acceptable for a moderate earthquake.

The distribution of flexural hinges for both designs is shown in Fig. 14. For the NEHRP design, nearly all the beams have formed flexural hinges, resulting in a very good SCWB design. However, it is disconcerting to note the extensive yielding that has occurred for only a moderate ground motion.

21 o Y.K. Wen et al.

w ~ ~ 0 , 0 2 0 . 0 1 I ' ' l ' ' ' l ' ' ' l ' ' ' l ' ' ' l ' ' ' l ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' ' 1 ' ' . . . . . . . . . ~ ~][[~l]~ dl l l l~l l ~ d l l l l ~

0.00 ~ ~ "~/~ .t~./\ ~ . . . . . . .

[ a -0.0!

- - 0 . 0 2 , I , , , I , , , 1 , . . I , , , i , , , I , , . I . , , i , , , I , . . I . . , [ . , L F * , , I , , , I , , 0 2 4 6 0 10 12 14 16 18 20 22 24 26 28 30

0.02 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

_ > f' . . . . . t

,~ 0.00 ~ A " A ", ,', ,, . . . . . . . t t - " ~ V ~ Z ~ . . . . ,°°'I

- - 0 . 0 2 ' ' l l a ' l ' ' ' l ' l f l ' ' ' l ' ' ' l l

0 2 4 6 8 10 12 14 16 111 20 22 24 I~0 20 30 l " ~ e ( .ec)

Fig. 13. Compar ison of global and local drifts for the NEHRP and dual-level designs subjected to the 1994 El Centro record.

The dual-level design, on the other hand, remained nearly elastic. It is interesting to note that if the El Centro record is taken as a model for the deterministic "design" earthquake, as many have done in the past, the NEHRP frame would be considered very well designed. However, the performance under the Sylmar record clearly demonstrates the NEHRP frame is not adequate and the E1 Centro record can no longer be considered a model for the deterministic "design" earthquake. Since the dual-level design remained nearly elastic, the overall damage index is only 0.02. Thus, no structural repairs should be needed for the dual-level design. The overall damage index for the NEHRP design is 0.26. This indicates that some moderate structural damage may need to be repaired after ground motions similar to the E1 Centro record. This is currently adequate performance for moderate ground motions, however, the incidental cost associated with repairing structural damage (e.g. profits lost from business closure, relocation of personal, etc.) have recently forced many engineers to consider stricter performance goals for moderate ground motions.

The Castaic-Old Ridge Route record of the 1971 San Fernando earthquake was chosen to represent a small earthquake. The acceleration time-history and response spectra are shown in Fig. 10. Both the NEHRP and dual-level designs remained elastic when subjected to the Castaic record. Thus, the

NEHRP Frame Dual Level Frame

Fig. 14. Compar ison of distribution of flexural binges of NEHRP and dual-level designs subjected to the 1940 El Centro

record.

Y.K. W e n e t al. 2 1 1

global drifts for both designs were relatively small (i.e. less than 0.25%). The interstory drifts remained well below the 0.5% threshold [1 ] for non-structural damage.

3.6. Reliability implications using 1994 Northridge earthquake records

The large number of strong ground motions recorded during the 1994 Northridge earthquake, provide an opportunity to evaluate the reliability of the NEHRP and dual-level designs. The 1994 Northridge earthquake occurred on a blind thrust fault dipping southward under the Santa Susana Mountains and the San Fernando Valley. In total, 84 corrected horizontal strong ground motion records from 42 stations have been used in this study to evaluate the reliability of the NEHRP and dual-level designs. Fig. 15 shows the spatial distribution of the stations over the Los Angeles basin area. The histogram of the epicentral distances of the stations is shown in Fig. 16 where the histogram of spatially uniformly distributed stations is also shown for comparison. Obviously near-source sites are over-represented in the records. The site soil conditions for each station could be approximated as stiff soil or rock, thus remaining consistent with the choice of soil factors for the design of the two frames. The Northridge records exhibited very high peak ground velocities, a better representation of

N

A "33

, |

uS ~9

m~

- . . . . . . . .-. " ' ~ ~

(

• 10 i

• 7 i

12

DIJ

I l l

"39

• 24 "23

"31

' ~ -- Epicenter ~

Fig. 15. Distribution of strong ground motion stations, 1994 Northridge Earthquake.

2 1 2 Y.K. Wen et al.

Spatially Uniform Distribution • Distribution of Northridge Records

0.2707

°24°1 n 0.2101

O. 180 t

I I o ~ 0.120

0.090

0.060

0.03

10 20 30 40 50 60 70 80 90100

Epicentral Distance (km)

Fig. 16. Histograms of epicentral distances of 42 recording stations of the Northridge earthquake.

[ ~ \.. 13 . . . . . . Dutl-Level Prime [ ~ A A . . . . 1'~ZalP I~r~ae

10-1 f ~ , , . ~ -

10 4

10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I~,~, 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Global Drift (~)

10 °

1 0 - 1

A

fl

lO,-m

10 ~ 0.0

. . . . I . . . . I . . . . I . . . . I . . . . I . . . .

~ . . [] l)u~-Level leremae N ~ . ~ /x . . . . N ~ ~'rtme

"-~ ~'-.a. ,,

, , , , I , , , , I . . . . I . . . . I i i i i I i , i

0.5 1.0 1.5 2.0 2.5 3.0 Loesl Drift (X)

Fig. 17. Probability of exceedance of global and local drift limits given the occurrence of Northridge-like earthquake within a reference area of 100 km radius.

Y.K. Wen et al.

Table 3 Comparison of global and local drift exceedance probabilities of NEHRP and dual-level designs

213

Global drift (0.5%) Global drift (1.5%) Local drift (0.5%) Local drift (2.0%)

NEHRP 0.125 0.006 0.20 0.018 Dual-level 0.075 0.003 0.15 0.006

the damage potential for the mid-period range. On the other hand, the peak ground displacements from the Northridge records were all relatively small, indicating that the Northridge earthquake did not seriously affect long period structures. The records exhibited the highest input energies of any California earthquake for natural periods less than 2.5 s, i.e. low or medium-rise building structures [21].

Probabilities were calculated of responses given the random occurrence of a Northridge-like earthquake in a reference area of radius of 100 km centered at the site. Probabilities are compared of overall flame damage index, global drift and local drift of the seven-story RC building according to NEHRP versus dual-level design procedure. Time-history responses to the 84 records were used as basis for the calculation. The effect of the non-uniform spatial distribution of the epicentral distance was considered and corrected. The responses to the actual records were then fitted by a proper extreme value distribution [22]. The probabilities of exceedance of global and local drift limits for the two flames are compared in Fig. 17 and Table 3. The probabilities for the dual-level flame are consistently below that for the NEHRP frame by a factor of about two, indicating that the dual-level flame is a better design if the engineer wishes to limit the global drift of the structure. The effect of the size of the reference area was also investigated and it was found that the area was large enough to provide a good estimate of the probabilities of exceedance. Note that although the above figures are conditional probabilities given the occurrence of the Northridge-like earthquake in the reference area. Since local events due to such a blind thrust fault are capable of causing significant loss to the society, they represent a severe test of the two design procedures and provide useful reliability information.

4. Dual-level design based on uniform hazard spectra

4.1. Background

In view of the many shortcomings in the current design procedures, an alternative method has been developed [23] which represents a significant departure from current procedures. The proposed procedure uses uniform hazard spectra to define the seismic hazard at a site and an equivalent system methodology to evaluate the performance of the structure. Performance is quantified in terms of the probability of exceeding displacement-based performance criteria. The parameters of the equivalent system methodology are calibrated using the results obtained from linear and nonlinear static pushover analyses of the structure. Deterministic design equations are developed based on the probabilistic performance criteria; these equations include "design factors" to account for the uncertainty in seismic hazard at the site, the uncertainty in predicting site soil effects, and the approximate nature of the equivalent system analysis methodology.

21 4 Y.K. Wen et al.

150.0 ' ' ' y . . . . , ' ~ ' ~ ~ , ~ ' ~ . ~ , ' ' ' I . . . .

1oo.o ~ ~ / / :'~'"-,,,, 50 .0

~ 0.0

-50.0

lOO0

~" coil

, oo : : - 150 .0 -100 .0 - 5 0 . 0 0.0 50.0 100.0 150.0

Distance from Site (km) Coastline ' ~ Site

Fig. 18. Seismic zones contributing to the seismic hazard at the site.

4.2. Uniform hazard spectra

Uniform hazard spectra provide a convenient means of incorporating probabilistic information into the design process. Each ordinate of a uniform hazard spectrum curve has the same probability of exceedance associated with it, and these curves can provide information on the elastic and inelastic response behavior of SDOF systems. The concept of using uniform hazard spectra in design is not new. Algermissen and Leyendecker [24] have suggested that approximate uniform hazard curves for elastic response can be constructed based on knowledge of the ordinates at two periods. This idea has been incorporated into the 1992 NEHRP provisions on a trial basis. Sewell and Cornell [25] have presented a technique for constructing uniform hazard spectra for nonlinear response. The basic idea of their approach is to scale elastic response ordinates by reduction factors which are dependent on the level of inelastic deformation, the frequency of the system, and other parameters. The ordinates of the resulting curves have equal probabilities of exceeding some measure of damage.

An alternative approach is to directly calculate uniform hazard spectra (for elastic and inelastic response) on a site-specific basis using simulation techniques. This approach is more computationally intensive, but it eliminates the need for empirical relations describing the variation of spectral reduction factors with damage level, period, etc. A site near Los Angeles, California is chosen for the study. The surrounding region is subdivided into "seismic zones" as shown in Fig. 18, and each seismic zone is assumed to have uniform seismicity characteristics. The seismic zones are based on the zones used by the US Geological Survey [26] in its seismic hazard studies of the region. Earthquakes are assumed to be exponentially distributed with respect to magnitude and inter occurrence time. In spite of the rather restrictive assumptions, the model is "sufficiently good" for engineering applications for the Los Angeles metropolitan area [27]. The site soil conditions are assumed to be approximately the conditions classified as S 2 in the NEHRP provisions. A total of

Y.K. Wen et al. 21 5

0.5

~0.4

0.3

~ 0.2

G P 0.1 ,.o

0.0

Prob. Level = 50X / 50 yearn

I eX • Elostic Cote / \ Op.=2 • l ip.:4.

T "r

1.0 2.0 3.0 4.0 0.0

1.0 , , ,

Prob. Level =, IOZ / 50 yeom

"~ • Elastic Case o" 0.8 / \ o~-2

I ~. ,,~.-4 o" 0.6 /

0.4

~ 0.2

0 . 0 ~

0.0 1.0 2.0 3.0 4.0

Fig. 19. Uniform hazard spectra of elastic and inelastic response. Each plot presents variation in the required force coefficient with period for a fixed probability of exceedance.

1292 simulated earthquake records were generated depending on the source, path and site condition. Elastic and inelastic responses of SDOF systems with various periods are calculated and from which the response spectra corresponding to a given probability of exceedance can be constructed. Examples of uniform hazard spectra generated using this approach in terms of fixed probability of exceedance for elastic and inelastic response of different ductility (/z) level are shown for Fig. 19. Note that the elastic displacement spectra S d (T) can be obtained by multiplying the force spectra by g (T/27r) 2, where g is the gravitation acceleration. The elastic spectrum compared well with that inferred from the 1992 USGS hazard map.

4.3. Equivalent SDOF system

Uniform hazard spectra such as those shown in Fig. 19 provide probabilistic information on the response of a SDOF oscillator. In order to use uniform hazard spectra in design, one needs to relate the response of a SDOF oscillator to the response of a MDOF structure. One method for doing this is to develop an "equivalent" SDOF model (also known as a generalized SDOF model) of the MDOF

21 6 Y.K. Wen et al.

1500.0

1250.0

1000.0

i 750.0

500.0

250.0

5.0 0 . 0 i i

0.0 10.0 15,0 20.0 Displacement (in)

Fig. 20. Typical lateral deflection profile at various stages of the push-over analysis.

system. Various procedures for doing this have been proposed in the literature [28]. The basic idea is to capture the displacement response of a MDOF structure in the inelastic range at some selected points on the structure. Fig. 20 show the displacement profile of a steel building under pushover test. The profile at a chosen deflection level (e.g. with a global drift ratio of 1%) is used to represent the MDOF structure under severe dynamic excitation. An equivalent SDOF system is then constructed using this deflection profile to derive the corresponding equivalent SDOF system parameters such as mass, stiffness, damping and participation factor. The original MDOF can be replaced by the equivalent SDOF system in the sense that (i) the global (roof) displacement is preserved; and (ii) the local (maximum interstory) drift can be obtained approximately via a conversion factor.

To validate the accuracy of the equivalent system methodology, the maximum roof displacement and maximum interstory drift ratio for six two-dimensional steel building frames were calculated for both real and simulated earthquake records, and the results were compared to the estimates obtained using the equivalent system methodology for the same records. The six building frames consist of a two-story moment-resisting frame (MRF), a nine-story MRF, a twelve-story MRF, an eight-story MRF with vertical irregularity in mass and stiffness, a five-story concentrically braced frame, and a five-story dual system consisting of a moment-resisting frame and a concentrically braced frame. For each frame, different equivalent system models were developed to predict linear elastic response and nonlinear inelastic response. For each frame static analyses were carried out using DRAIN-2DX. The building frame models and the equivalent system models were analyzed for a set of "small to moderate" earthquake records and a set of " l a rge" earthquake records. The dynamic analyses were carried out again using DRAIN-2DX. Both real and simulated records were used. For the nonlinear equivalent systems, the smooth hysteresis model [29] was used. Fig. 21 shows a set of four scatter plots which compare the bias factor for a particular response quantity (displacement or drift ratio), the ratio of the response calculated using the MDOF model to the response calculated using the equivalent system model. The mean values of the bias factor for roof displacement are all within the range 0.73 to 1.03. The mean values of the bias factor for drift ratio range from 0.67 to 1.44. Also, there is more scatter in the data for drift. The higher scatter is to be expected since the basic equivalent system formulation is for roof displacement, and the conversion of roof displacement to maximum interstory drift ratio is very approximate.

It is pointed out that simplified analyses using equivalent system models of MDOF building frames should not be expected to provide accurate results under all conditions. Such analyses nevertheless

Y.K. Wen et al. 2 1 7

.~ 60.0 II "-N

50.0 I

E 0 0 _o

o

b_ o o

40.0

30.0

20.0

10.0

O.O 0.0

0 , i O 0 ' '

• Real record / , , [ • Real ,r~cord,

10.0 20.0 30.0 40.0 50.0 60.0 MDOF Max. Roof Displacement (in)

I

#. O r ¢

"C a

O O

la.

8

5.0

4.0

3.0

2.0

1.0

0.0 0.0

i i i i

, , ~ - I 0 Simulated record

1.0 2.0 3.0 4.0 5.0 MDOF Max. Interstory DHft Ratio (X)

I

0 E ¢)

13. ._m C3

0 :Z b . 0 0

60.0

50.0

40.0

30.0

20.0

10.0

0.0 0.0

i i ! O i i

oY / ~ ' - I o Slmulat~l record

10.0 20.0 30.0 40.0 50.0 60.0 MDOF Max. Roof Displacement (in)

NONUNEAR

-- 5.0

I 4.0

0 ~ 3.0

," 2.0 IP

E 1.0 8 h

o ~ 0.0

0

o

0.0 1.0 2.0 3.0 4.0 5.0 MDOFMox. lntereto~ D R Ratio(Z)

RESPONSE

Fig. 21. Scatterplot comparing maximum roof displacement and interstory drift predicted by equivalent SDOF models with nonlinear responses obtained from a MDOF model of a 12-story MRF.

can provide valuable information for structural design, as long as the approximation and associated error is properly accounted for in design. Several issues related to the equivalent system methodology, however, must be addressed before it is implemented into design applications. First, the proper choice of the lateral force distribution must be determined. Second, the results of this study are based on two-dimensional analysis models of building frames. Such models are acceptable if torsional effects are small and if three-dimensional effects can be accounted for separately [7]. Finally, additional research is needed to develop simple, yet realistic, methods of modeling the nonlinear behavior of MDOF structures for nonlinear push-over analyses.

4.4. Design format and consideration of uncertainty

In the proposed dual-level procedure, the two stages of design and analysis correspond to the two limit states to be considered (serviceability and ultimate). The proposed procedure assumes that

2 1 8 Y.K. Wen et al.

0 . 5 , t , I i

. . . . . . . v~t ,=290 m / s

. . . . . . . . . . . v ,~ ,=540 m / s F.-1.8 v=a.= 1050 m / s

0.4 . . . . . . . . . . . . . • ,,,, v,~= 1620 m / s

F.--1.3 " , \ . . . . . . . . . . -% •

o." ' \ ""'..r,-2 3 v n ~ = 1050 m/s F.= I .0 " ' -

,?.=

~ o.2 ~ ~ .... ... F,-1.5 . . . . . . . . . .

o . , ~ : - ~ . ~ " - - ~ _ . . . . . . . . . .

0 . 0 I I I I I

0.0 0.5 1.0 1.5 2.0 2.5 3.0 Pedod (ram)

Fig. 22. Design spectra based on B orcherdt's (1995) empirical soil amplification factor as function of soil mean shear wave velocity.

interstory drift limits and the corresponding target probabilities of exceedance are specified by future codes for both limit states. Furthermore, for the ultimate limit state, it is assumed that the codes limit the maximum global ductility ratio. The global ductility ratio is defined as the maximum roof displacement divided by the global yield displacement; the global yield displacement is determined from a static nonlinear pushover analysis of the structure. This latter requirement is used in lieu of response reduction factors (R factors) to determine the strength required to limit inelastic deforma- tion. With this criteria, the nonlinear behavior of the structure is explicitly (albeit approximately) considered in design. Soil amplification of ground excitation plays an important role in seismic response analysis and design as indicated by field evidence collected in recent earthquakes such as 1989 Loma Prieta earthquake. Recent research [30] indicated that the spectral amplification due to soil can be modeled empirically by a soil factor as a function of the mean soil shear wave velocity:

)m(Aa,T) Uref

f = - - , ( 6 ) Usite /

where /)re f is the mean shear wave velocity for the reference soil conditions; Usite is the mean shear wave velocity for the soil profile at the site; and m( A a, T) is an exponent which is a function of the period, T, of the structure and the severity of the input ground motion described by the parameter Aa, the effective peak ground acceleration. Fig. 22 shows the design spectra based on this formula. Significant amount of uncertainty (with a coefficient of variation of 0.50), however, exists which needs to be taken into consideration in design. Fig. 23 shows the comparison of the predicted soil factor with observed data from Loma Prieta Earthquake. In the proposed procedure, ground excitation uncertainties have been accounted for in the uniform hazard spectra; the uncertainties in soil amplification and structural modeling are included in the design format through the use of "design factors" which are similar in concept to the LRFD method. The reliability analysis required for the

Y.K. Wen et al. 2 1 9

b -

0 3

" C

I 1g o

0 3

10 1

10 0

I I I I I I I I

v,~. ,o . = 1050 m / s

o oooo o o

o o

o o 0 o o

0 Raw Data Proposed Formula

1 0 - 1 I J I J t I t I

10 10 3

Meon Shear Wave Velocity, v,~ (meters/second)

Fig. 23. Comparison of predicted soil factor with observed data from 1989 Loma Prieta earthquake.

derivation of the design format can be found in [23]. The resulting design procedure is outlined in the following.

At the serviceability level, the performance criterion is that the structure remains linear elastic and the probability of interstory drift ratio, A, exceeding the code specified level, A S, over a given time interval (e.g. 50 years) be less than Ps:

e ( a > as)-<Ps. (7) The corresponding deterministic design-checking equation is:

HAs Sd(T*) -< J'2tnfP *B ' (8)

in which S d = elastic spectral displacement obtained from the uniform hazard curve corresponding to target probability Ps; T * = period of the equivalent SDOF system; H = total structural height; n = structural model bias factor; f = site soil factor calculated using Eq. (6); P * and B are equivalent system participation factor and interstory drift factor. ~ t is the design factor which accounts for the fact that the soil factor, the structural modeling error, and the elastic spectral displacement are all random variables. /2 t can be calibrated for a given target exceedance probability Ps. Table 4 shows values of O t and the design elastic force coefficient C e for different structural periods and a target Ps of 50% in 50 years and 10% in 50 years. Notice the weak dependence of Ot on period and target reliability level since influence of these two factors are accounted for primarily by C~.

At the ultimate level, the first performance criterion is that the probability of structural global ductility /z, exceeding a prescribed level, /x u, over a given time interval be less than Pd:

P(/-t >_/-tu) _< P~I, (9)

and the corresponding deterministic design-checking equation is:

Dy >_ [2tFP * fg ~ Cy(]"£t),

tO

0

Tab

le 4

D

esig

n va

lues

of

C c

and

g2 t

for

elas

tic

resp

onse

as

func

tion

of

targ

et e

xcee

danc

e pr

obab

ilit

y, s

truc

tura

l pe

riod

and

sit

e so

il c

ondi

tion

Per

iod

(sec

) O

'log F

D

esig

n va

lue

of C

e D

esig

n fa

ctor

con

side

ring

F =

F

(A a

, v)

for

var

ious

val

ues

of s

ite

She

ar w

ave

velo

city

, v

a,b

v =

1620

v

~ 10

50

v =

v~e f

= 54

0 v

= 29

0 v

= 15

0

Tar

get

exce

edan

ce p

roba

bili

ty (

Pt)

= 5

0% i

n 50

yea

rs

0.1

0.21

0.

24

1.26

1.

26

1.25

1.

25

1.25

0.

3 0.

21

0.49

1.

28

1.28

1.

28

1.28

1.

27

0.5

0.18

0.

42

1.24

1.

24

1.24

1.

24

1.24

0.

7 0.

18

0.33

1.

19

1.19

1.

19

1.18

1.

18

1.0

0.18

0.

26

1.15

1.

14

1.14

1.

14

1.13

2.

0 0.

18

0.13

1.

16

1.16

1.

15

1.15

1.

14

3.0

0.18

0.

091

1.14

1.

14

1.13

1.

13

1.12

Tar

get

exce

edan

ce p

roba

bili

ty (

Pt)

z

10%

in

50 y

ears

0.

1 0.

21

0.46

1.

56

1.42

1.

29

1.21

1.

17

0.3

0.21

0.

91

1.57

1.

43

1.31

1.

25

1.20

0.

5 0.

18

0.71

1.

35

1.31

1.

26

1.24

1.

22

0.7

0.18

0.

62

1.32

1.

27

1.21

1.

16

l.l

3 1.

0 0.

18

0.49

1.

31

1.26

1.

19

1.14

1.

10

2.0

0.18

0.

27

1.28

1.

23

1.15

1.

10

1.06

3.

0 0.

18

0.19

1.

28

1.23

1.

15

1.10

1.

05

a T

he v

aria

tion

of

F fo

r A

a v

alue

s be

twee

n 0.

1 an

d 0.

4 is

con

side

red.

b

She

ar w

ave

velo

city

val

ues

are

give

n in

uni

ts o

f m

/sec

.

Tab

le 5

D

esig

n va

lues

of

Cy

and

fact

or g

2 ff

as f

unct

ions

of

targ

et e

xcee

danc

e pr

obab

ilit

y, s

truc

tura

l pe

riod

and

tar

get

duct

ilit

y

Per

iod

(sec

) O

'~o~

F

Tar

get

duct

ilit

y =

2

Tar

get

duct

ilit

y =

4

Tar

get

duct

ilit

y =

6

Val

ue o

f C

y (/

~ =

2)

from

UH

S

~F

V

alue

of

Cy

( ~£

=

4) f

rom

UH

S

Ot r

V

alue

of

Cy

( ~

= 6)

fro

m U

HS

g2

/

Tar

get

exce

edan

ce p

roba

bili

ty (

Pt)

= 1

0% i

n 50

yea

rs

0.1

0.21

0.

32

1.4

3

0.26

1.

41

0.23

1.

42

0.3

0.21

0.

42

1.4

5

0.26

1.

40

0.20

1.

39

0.5

0.18

0.

33

1.3

2

0.20

1

.27

0.

14

1.27

0.

7 0.

18

0.27

1

.29

0.

15

1.26

0.

11

1.27

1.

0 0.

18

0.21

1

.27

0.

12

1.2

5

0.08

0 1.

29

2.0

0.18

0.

12

1.3

0

0.06

0 1

.28

0.

043

1.32

3.

0 0.

18

0.07

8 1

.25

0

.04

1

1.2

2

0.02

7 1.

27

e~

Tar

get

exce

edan

ce p

roba

bili

ty (

Pt)

= 10

% i

n 10

0 ye

ars

0.1

0.21

0.

40

1.4

3

0.33

1

.42

0.

29

1.43

0.

3 0.

21

0.52

1

.46

0.

32

1.40

0.

25

1.40

0.

5 0.

18

0.41

1

.32

0.

25

1.2

7

0.18

1.

27

0.7

0.18

0.

34

1.2

9

0.20

1

.26

0.

14

1.27

1.

0 0.

18

0.27

1

.27

0.

16

1.2

5

0.10

1.

30

2.0

0.18

0.

15

1.3

0

0.07

7 1

.28

0.

053

1.31

3.

0 0.

18

0.10

1

.25

0.

056

1.21

0.

034

1.28

t,3

1",3

222 Y.K. Wen et aL

where /Xu

] A t = /z D I S P ' (10) ~"~t n u

in which Dy is global yield displacement; T *, P * and f have been previously defined; g is the acceleration of gravity. /2 F is the design factor accounting for the variability in the soil factor; Cy(/x t) is the yield force coefficient at T = T * obtained from the uniform hazard spectrum curve corresponding to ductility /x t and probability pl. ~t" is the design factor accounting for the random nature of the equivalent system bias factor NDISP; n D/sP is the mean bias factor. Table 5 presents the values of the design factor ~ t F for two target probabilities of exceedance and four target ductilities. It is used in conjunction with the design yield force coefficient, Cy(/xt), and the design site soil factor, f. In this study, the design value of Cy(/x t) is chosen to be the value for which the probability of exceeding /x u is equal to the target probability p~ under consideration. It is seen that the value of g2t F is period-dependent. The same type of period-dependence was observed for the design factor discussed earlier in connection with linear elastic response. Also, as is the case for /2t , /2t F does not vary significantly with target exceedance probability or structural period since these two factors are accounted for by Cy(/xt). For a fixed period, the design factor does not, in general, vary significantly for ductility ranging from 2 to 6. The maximum variation is about 6%. This is significant for code implementation since it implies that a single value of the design factor can be specified at each period which is valid for a wide range of ductility.

The second performance criterion is that the probability of maximum local (interstory) drift, A, exceeding a prescribed level, Au, over a given time interval be less than P~:

P(A@Dy> Au)<_p 2. (11)

The corresponding deterministic design-checking equation is: B

__ " , ~'~ anDRIFT D A u > / X ~ t u y, (12)

where g2t a is the design factor accounting for the random nature of the equivalent system bias factor NI~RIFT; n DRIFt is the mean bias factor; /x' is a ductility corresponding to Dy. To calibrate values of J2t" and Et a for both N DISP and N DR~r, respectively, information on the probability distribution functions for ductility at fixed values of yield displacement are required. Due to a lack of adequate information on these distribution functions, values of /?t" and ~ta were not calculated. However, based on some very approximate calculations, it appears that, in general, the values of g~t" and g~t a range from 1.1 to 1.4. The values of both of these design factors would be expected to be smaller than the values of g2 F since the values of the coefficient of variation for the bias factors are on the order of 20% and the coefficient of variation in the site soil factor is 40-50%.

Iterative design procedures were also developed based on the above dual-level probabilistic design criteria and corresponding deterministic design-checking equations. The procedures have been demonstrated on the design of a two-story steel frame building [23].

5. Summary and conclusions

Current code procedures for design against seismic forces have been largely based on experience of performance of structures in past earthquakes. The large uncertainty associated with the excitation and

Y.K. Wen et aL 223

inelastic structural response behavior have not been fully accounted for. To achieve risk and performance-based design goals, three dual-level design methods with consideration of the service- ability and ultimate limit states are proposed. The first is a method of calibration of design parameters according to the two target reliability levels within the context of current code procedures. The method is demonstrated by calibration of seismic load factor and drift limit. The second is an extension of the design earthquake approach to allow consideration of two levels of design earthquake. The performance is assured by linear and nonlinear equivalent static (pushover) analyses. Comparison of structural performances based on this approach and conventional method indicated that the dual-level procedure produces superior designs which come closer to the design objectives of most current codes. The third is an alternative to the current design procedures. It uses uniform hazard response spectra and a pushover analysis-based equivalent single-degree-of-freedom system to explicitly account for variability and uncertainty in the seismic loads and inelastic response behavior of the structures. This avoids the use of the controversial response reduction factor. Uncertainties in the soil amplification and structural system modeling are accounted for by corresponding design factors similar in concept to the LRFD method. It ensures that the structure meets local (interstory) drift and global ductility requirements with specified probabilities.

Acknowledgements

This study is supported by the National Science Foundation under Grants NSF BSC-9106390, NSF CMS-9415726, and NSF NCEER-934102B. The support is gratefully acknowledged.

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