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CAPACITY-DEMAND CURVES METHOD FOR PERFORMANCE/

DISPLACEMENT-BASED SEISMIC DESIGN

L. P. Ye1

Department of Civil Engineering Tsinghua University

Beijing P. R. China 100084 [email protected]

Abstract

Based on the visual graphical procedure of capacity spectrum method (CSM), a new method, called capacity-demand curves method (CDCM), is proposed for performance/displacement seismic design in this paper. The new method can be drawn from CSM but could avoid its disadvantages in difficult determination of demand point and larger error induced for long period system. Then the method is developed to determine the seismic performance/displacement including damage degree of structures and the corresponding probability under multiple earthquake hazard levels. All the procedures are presented in illustrative format which are easy to be understood and the values are easy to be determined. The demand curves, reflecting the relations of elastic and inelastic response, are based on the suggestion of R-µd relations in previous studies, and comparison of different models is discussed.

1 Introduction

In recent year, studies on performance/displacement-based seismic design method were taking out around the researchers of earthquake engineering in the world. The seismic design method innovation is promoted for that designer want to evaluate and verify the structures (existing or to be designed) performance under multiple earthquake hazard levels. Some evaluation and design procedures incorporating with performance/displacement-based seismic design concept were developed in some requirements and documents, such as Vision 20001, ATC-402 and new seismic design provision in Japan3. Among them, the so-called capacity spectrum method (CSM), which compares the capacity spectrum of structure with demand spectrum of earthquake ground motion using visual graphical procedure, was become popularity. This visual graphical procedure is easy to be understood for the seismic performance of the structures existing or to be designed.

The CSM (see Fig. 1) was developed by Freeman4,5, and some modifications both in determining capacity spectrum of structure and demand spectrum of earthquake ground motion were made in recent years6,7,8. Besides the problems to obtain more reasonable and accuracy of the both spectrum curves, CSM has two other following disadvantages in practical use,


(1) The demand point must be determined by the intersection of the capacity spectrum and the demand spectrum with corresponding damping factor or ductility factor. The inelastic seismic demand spectrum are usually provided in a set of damping factors or ductility factors, so it makes some trouble to find the proper demand spectrum corresponding to the demand point at the capacity spectrum.

(2) For long period structure, the resistance strength of structure capacity curve is usually small, which results in a small value of capacity acceleration in the capacity spectrum. It may cause a large error in finding the demand point due to flat demand spectrum curves in long period range.

But the visual graphical procedure of capacity spectrum method is attractive in determining the seismic performance. Thus, the author proposed a new visual graphical procedure, called capacity-demand curves method (CDCM), for performance/displacement seismic design in this paper. The new method, without the above two disadvantages of CSM, is firstly drawn from CSM to present its concept. Then the method is developed to determine the seismic performance of structures under multiple earthquake hazard levels. All the procedures are presented in illustrative format which are easy to be understood. Finally, demand curves in CDCM of different models are discussed.

2 CONCEPT OF CAPACITY-DEMAND CURVES METHOD

Fig. 2 shows CSM used for a serials of structure system with same initial period T . The straight line OA in Fig. 2 represents the elastic capacity spectrum of the structure equivalent SDOF system (‘the structure equivalent SDOF system’ will be omitted when ‘structure capacity spectrum’ is used in the paper following). Line OA and elastic demand spectrum intersect at point A, which is the seismic demand for the elastic system. The spectral acceleration of point A is taken as SA,E.

Elasto-plastic capacity spectrum of structure is supposed to be used for inelastic system. The yielding spectral acceleration of the inelastic system capacity spectrum is expressed by SA,E/R. The factor R is called strength reduction factor due to inelastic properties of inelastic system, such as ductility and hysteretic energy dissipation. With a definite value R, say R1 or R2, a corresponding capacity spectrum curve can be obtained, see Fig, 2. As for the capacity spectrum corresponding to factor R1, the demand point (point B in Fig. 2) can be determined using CSM. And another value R2 corresponds to another demand point (point C in Fig. 2). In this way, a demand curve (i.e., the curve ABC in Fig. 2), can be obtained for serials of reduction factor by connecting the demand points. If the demand curve can be given before hand for any initial period (Fig. 3), the intersection of the capacity spectrum and the

Figure 1 Capacity Spectrum Method

SA

SD

Demand spectrum corresponding to point A

Realistic response point Capacity spectrum #1

Demand spectrum corresponding to a set of damping factors or ductility factors

Capacity spectrum #2

0

A


demand curve (notice the difference between demand curve and demand spectrum) can be determined to obtain the inelastic response without interpolation calculation as CSM.

In a more comprehensive illustration, the capacity spectrum and the demand curve can be expressed in force-displacement coordinate system, as shown in Fig. 4. Actually, the capacity spectrum in acceleration-displacement coordinate system is converted from the capacity curve of structure in force-displacement coordinate system by dividing the force by the effective mass of equivalent SDOF system5,7. Thus, the capacity curve and the demand curve can be used to determine the seismic response point of an inelastic system by illustration procedure in force-displacement coordinate system, so the method is called capacity-demand curves method (CDCM). Because the demand curve across the capacity curve around 90 degree angle and a proper scale of force-displacement coordinate system can be used for a special initial period in the CDCM, the intersection point between the two curves is easy to be determined for any initial period system. Thus, the two disadvantages stated previously can be overcome.

Figure 2 Concept of Demand Curve Based on CSM

SA

SD

Demand spectrum corresponding to point C Demand point of Capacity spectrum of R2 Capacity spectrum of R2

Elastic demand spectrum

0

C

Elastic response point

Demand spectrum corresponding to point B

Demand point of Capacity spectrum of R1 Capacity spectrum of R1

SA,E SA,E/R1

SA,E/R2

B

Demand curveT

A

SA

SD

Elastic demand spectrum

0

Demand curve for T1

T1

Figure 3

Demand curve for T2

T2

Demand curve for T3

T3


From previous description, the suggested CDCM not only has the advantages in CSM with a whole performance evaluation in spectrum way (Fig. 3), but also can be used for a special case of initial period (Fig. 4, this expression of the CDCM will be used in following of this paper). With the elastic response point being determined from the elastic demand spectrum and the known corresponding demand curve, the performance/displacement-based seismic design procedure, stated by Fajfar7, can be also adopted in the CDCM. Fajfar7 has also pointed out that ‘the inelastic seismic demand can be determined without constructing the demand spectra’. The determination of demand curves will be discussed later in this paper.

3 PERFORMANCE UNDER MULTIPLE EARTHQUAKE HAZARD LEVELS

The structure performance should be evaluated under multiple earthquake hazard levels in the performance/displacement-based seismic design. This can also be made by the CDCM in an easy way.

Three earthquake hazard levels, representing small, moderate and severe earthquake intensities, are considered. The three demand curves corresponding to the three levels are assumed proportional to the elastic responses, as shown in Fig. 5. Thus, the intersections of capacity curve and the demand curves give the performance estimation under the three earthquake hazard levels.

In Fig. 5, the yield strength of capacity curve is shown larger than the elastic force response at small earthquake level. This is essential for structure within elastic range and obvious damage is not allowed under small earthquake, so the demand curve of small earthquake is not shown. From the intersections of capacity curve and the demand curves under moderate and severe levels, the damage degree of structure can be estimated as,

cu

dDM,µ

µ= (1)

where, DM is damage degree value of structure, DM=0 represents no damage and DM=1 represents total damaged; µd is the demand ductility factor, µd=Dd/Dy; Dd is the demand displacement (Dd,M or Dd,L in Fig. 5 for moderate and severe earthquake intensity), which is determined by the intersection of capacity curve and demand curve; Dy is the yield displacement of the equivalent SDOF system of structure; µu,c is the ultimate ductility factor under cyclic (earthquake) loading, which can be determined by Fajfar’s suggestion9 that µu,c can be obtained from the ultimate ductility factor µu,m under monotonic loading by considering hysteretic

Force

Displacement 0

Demand curve

K

Figure 4 CDCM in Force-Displacement Coordinate System

Elastic response

Inelastic responseCapacity curveCapacity limit


energy accumulation under ground oscillation. If the damage degree limits are given for moderate and severe earthquake levels, it is easy to determine the yield strength Fy by the CDCM.

4 PROBABILITY OF STRUCTURE PERFORMANCE BY CDCM

Another important value that should be estimated in performance/displacement -based design is the reliable (or lose) probability of the structure performance. Supposing that the exceeding probability of an certain earthquake intensity at the building site is known by the history earthquake investigation and earthquake risk analysis, the damage probability of the structure can also be estimated by the CDCM.

Fig. 6(a) shows probable frequency curve of earthquake intensity. The horizontal axis is earthquake intensity, IS, IM and IL represent the small, moderate and larger (severe) earthquake intensity respectively. The vertical axis is probable frequency corresponding to the earthquake intensity. The exceeding probability of a earthquake intensity, say moderate earthquake intensity IM, is the area under the probable frequency curve at the right of IM (shade area in Fig. 6(a)), it can be calculated by Eqn. (2). The exceeding probability curve is shown in Fig. 6(b).

∫∞

==

IxdxxpIP )()( (2)

As the elastic response is approximately proportional to earthquake intensity, so the exceeding probability of elastic response and then the corresponding demand curve is same as that of the earthquake intensity, for example, P(Fe,S)=P(IS), P(Fe,M)=P(IM) and P(Fe,L)=P(IL), etc. Thus, considering a continuos earthquake intensities, the exceeding probability curve of damage degree for a given capacity curve, saying capacity curve #1 in figure 7, can be determined used the illustration procedure as shown in figure 7. By this way, the structure damage probability under the potential earthquake intensities can be evaluated. In figure 7, it is easy to understand that the larger yield strength of capacity curve with the same ultimate ductility capacity, the less exceeding probability of damage

Note: Point S, M and L represent elastic responses under small, moderate and severe earthquake levels; Dd,M and Dd,L represent demand displacements of moderate and severe earthquakes, respectively; Fe,S, Fe,M and Fe,L represent elastic force responses of small, moderate and severe earthquakes, respectively.

Figure 5 Performance under Multiple Earthquake Hazard Levels

Force

Displacement 0

Demand curve of severe level

K

MCapacity curve

L

S Fy

Dy Dd,M Dd,L

Demand curve of moderate level

Fe,L

Fe,M

Fe,S


degree. Thus, designer can make a proper decision for the designed structure with an expecting damage degree probability.

p(I)

Earthquake intensity II IM IS

Exceeding probability of IM

0

(a) Probable frequency curve

Figure 6 Probability of earthquake intensity

P(I)

Earthquake intensity IL IM IS

P(IM)

0

(b) Exceeding Probability curve

P(IS)

P(IL)

1.0

Force

Displacement 0

Demand curve of severe level

K

M

Capacity curve #1

L

S

Dy Dd,M Dd,L

Demand curve of moderate level

P(Fe,L)=P(IL)

Probability 1.0

P(Fe,M)=P(IM)

P(Fe,S)=P(IS)

(a) Relation between demand curve and probability of earthquake intensity

Capacity curve #2


In above procedure, because the demand curve is the average one for certain earthquake intensity, the exceeding probability is corresponding to the average demand response. The response randomness under certain earthquake intensity should also be considered in determining the exceeding probability in future study.

5 DEMAND CURVES

The demand curves are in fact the response relations of inelastic system and elastic system. For the equivalent SDOF system of structures and an elasto-plastic force-displacement relation being used, the demand curves can be expressed as,

ed

d DR

Dµ

= (3)

where, Dd is the demand displacement of inelastic system for an earthquake intensity; De is the elastic displacement response under the same earthquake intensity; R is strength reduction factor, R=Fe/Fy; Fe is the elastic force response under the same earthquake intensity; Fy is the yield strength of the inelastic system. If the relations between the strength reduction factor R and the demand ductility µd are known, the demand curves can be obtained.

The Newmark and Hall’s10,11 equal energy and equal displacement rules may be regarded as the first and simple R-µd relations suggestion. But Newmark and Hall’s suggestion did not consider the continuous influence of initial period of the systems and sometime underestimate inelastic response. Several following researches12-19, including the method of equivalent damped linear elastic system of inelastic system13, were made on establishing the R-µd relations. An excellent and thorough review of the previous efforts before 1994 on the relations can be found in the work by Miranda and Bertero16.

Based on the previous researches, R-µd relation depends on the initial period T of the systems, hysteresis energy dissipation capacity and site types. Recently, Ordaz17 found that elastic displacement spectrum had relations to the strength reduction factor R, and so to the demand curves. But Ordaz’s suggestion can only be used for individual earthquake motion with its elastic displacement spectrum or maximum displacement of ground motion given. Akiyama18 and the author of this paper19 deduced the relations based on energy concept. The author has also discussed the continuous influence of initial period of systems based on energy input spectra. In those studies, the energy-based approach may be the one to give a reasonable theory explanation about the response relation between inelastic systems and elastic systems, while the others were mainly based on response properties and statistics analysis.

Figure 7

(b) Probability of damage degree

P(DM)

DM0

Probability curve of Capacity curve #2

0.5 1.0

Probability curve of Capacity curve #1

P(DM(µd,M))=P(IM)

P(DM(µd,L))=P(IL)


The R-µd relations suggested by the previous researches are not presented in detail in this paper, but Fig. 8 shows their comparison in 1/R–Dd/De relation (same pattern as the demand curves) for short and middle initial period systems. It could be noticed in Fig. 8(a) that Newmark’s equal energy rule appear to be the average of the other researchers’ suggestion in short period field, this is due to that other models chosed a proper initial period parameter. It could also be noticed in Fig. 8(b) that Newmark’s equal displacement rule is on the safe side in middle period field when strength reduction factor R is larger than a reasonable low value, and the Akiyama’s suggestion is shown to be the lower bound because it toke a perfect elasto-plastic

0 0. 1 1. 2 2. 3

0.

0.

0.

0.

1 Newmark equal energy rule

Vidic, /TG=0.5Miranda, T=0.25s(rock site)Shibata, <TG

Ye, T<TG, ζ=0.6Akiyama, T<TG,Ordaz, /Dmax=0.3 Nassar, =0.25s

Dd/De

(a) In short period field

1/R

0 0. 1 1. 2 2. 3

0.

0.

0.

0.

1 Newmark equal displ. ruleVidic, >TG

Miranda, T=2s(rock site)Shibata, T>TG

Ye, T>TG,Akiyama, T>TG,Ordaz, D/Dmax=1.5Nassar, T=2s

Dd/De

(b) In middle period field

1/R

Figure 8 Demand Curves of Different Models


hysteresis model.

Another thing the author should mention is that calculated results with inelastic time history analysis method were found larger than any demand curves suggestions in cases, especially for low strength reduction factor in short period field19. Thus, the seismic response randomness and the reliable probability of demand curves suggestion need further study. Nevertheless, the suggestion of R-µd relations in previous studies can provide reasonably accurate both for the CDCM and capacity spectrum method for seismic design situations.

6 CONCLUSIONS

With the merits of visual graphical procedure and without the disadvantages of difficult determination of demand point and larger error may induced in long period by CSM, the CDCM proposed in this paper has shown to be a more useful method for performance/displacement-based seismic design for new built structure and evaluation of existing buildings. The CDCM can also be used for evaluating the performance and damage degree of structures under multiple earthquake levels. If the exceeding probability of earthquake intensities is known, the reliability probability of performance and exceeding probability of damage degree of structures can also be evaluated by the CDCM in illustration format. The accurate of the CDCM is mainly based on the demand curves, which many researches had been done. The reliable probability of demand curves due to the seismic response randomness is worthwhile to investigate in further study both for CDCM and CSM. But the suggestion of R-µd relations in previous studies can provide reasonably accurate both for CDCM and CSM for seismic design and evaluation.

REFERENCES

[1] SEAOC Vision 2000 Committee(1995). Performance-based Seismic Engineering. Report Prepared by Structural Engineering Association of California, Sacramento, California, U.S.A.

[2] ATC(1996). Seismic evaluation and retrofit of concrete buildings. Vol.1, ATC-40, Applied Technology Council, Redwood City.

[3] S. Otani, H. Hiraishi, etc(2001), New Seismic Design Provisions in Japan. Proc. of International Conference on Advanced Technologies in Design, Construction and Maintenance of Concrete Structures, Hanoi, 1-10

[4] S. A. Freeman, J. P. Nicoletti and J. V. Tyrell(1975). Evaluations of exiting buildings for seismic risk-A case study of Puget Sound Naval Shipyard, Bremerton. Washington. Proc. 1st U.S. National Conf. Earthquake Engng., EERI, Berkeley, 113-122.

[5] S. A. Freeman(1998). Development and use of capacity spectrum method. Proc. 6th U.S. National Conf. Earthquake Engng., Seattle, CD-ROM, EERI, Okland,.

[6] H. Krawinkler and G. D. P. K. Seneviratna(1998). Pros and cons of a pushover analysis for seismic performance evaluation. Engng. Struct. 20. 452-464.

[7] P. Fajfar(1999). Capacity spectrum method based on inelastic demand spectra. Earthquake Engng. Struct. Dyn. 28, 979-999.

[8] V. V. Bertero(1992). Tri-service manual methods. in Vision 2000, Part 2, Appendix J, Structural Engineering Association of California, Sacramento, California, U.S.A, 1995.

[9] P. Fajfar(1997). Equivalent Ductility Factor Taking into Account Low-cyccle Fatigue’, Earthquake Engng. Struct. Dyn., 21, 837-848.

[10 ] A. S. Veletsos and N. M. Newmark(1960). Effect of Inelastic Behavior on the Response of SimpleSystem to Earthquake Motion. Proc. of Second World Conference on Earthquake


Engineering, 895-912.

[11] N. M. Newmark and W. J. Hall(1982). Earthquake Spectra and Design, Earthquake Engineering Research Institute, Berkeley.

[12] A. Nassar and H. Krawinkler(1991). Seismic demands for SDOF and MDOF systems. Report 95, The John A. Blumn Earthquake Engineering Center, Stanford University

[13] A. Shibata(1975). Study on inelastic response of nonlinear structures for earthquake motion by equivalent linear system method’, Report of Northeast University, No.16, (in Japanese)

[14] T. Vidic, P. Fajfar and M. Fischinger(1994). Consistent inelastic design spectra: strength and displacement’, Earthquake Engng. Struct. Dyn. 23, 502-521.

[15 ]E. Miranda(1993). Site-dependent strength reduction factots. J. Struct. Engng. ASCE, 119(12), 3503-3519.

[16] E. Miranda and V. Bertero(1994). Evaluation of strength reduction factor for earthquake-resistant design. Earthquake Spectra 10(2), 357-379.

[17] M. Ordaz and L.E. Perez-Rocha(1998). Estimation of strength-reduction factors for elastoplastic systems: a new approch, Earthquake Engng. Struct. Dyn. 27, 99-901.

[18] H. Akiyama, Earthquake-resistant limit-state design for buildings, university of Tokyo Press, 1985

[19] L. P., Ye and S. Otani(1999). Maximum seismic displacement of inelastic systems based on energy concept. Earthquake Engng. Struct. Dyn. 28, 1483-1499

[20] H. L. Li, W. H. Sang and H. O. Young(1999). Determination of ductility factor considering different hysteretic medels. Earthquake Engng. Struct. Dyn. 28, 957-977

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