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PRE-PUBLICATION COPY 10/30/95

A REVIEW OF PRACTICAL APPROXIMATE INELASTIC SEISMIC DESIGN PROCEDURES FOR NEW AND EXISTING BUILDINGS

Sigmund A. Freeman1

ABSTRACT

Currently, there are several projects that are developing performance based seismic provisions that are based on approximate inelastic design procedures. These projects include: 1. FEMA Project ATC-33, Guidelines and Commentary for the Seismic Rehabilitation of Buildings; 2. State of California Proposition 122 Project ATC-40. Recommended Methodology for Seismic Evaluation and Retrofit of Existing Concrete Buildings; 3. SEAOC Project Vision 2000; 4. the TriServices update of TM 5-809-10-1, Seismic Design Guidelines for Essential Buildings; and. 5. the City of Los Angeles proposal for upgrading masonry infilled concrete frame buildings.

The basis for the proposed methodologies have· been around for a long time in various forms. Now that there is apparently a serious effort to develop approximate inelastic procedures for practical use in seismic design provisions. they are being widely reviewed, critiqued, and formalized.

The purpose of this paper is to review the proposed methodologies in a manner than can be understood by the design profession. Also to be discussed are such terms as the "equal displacement rule," "inelastic response spectra". "pushover analysis," "secant modulus methods." "substitute structure." and "capacity spectrum method."

1Principal, Wiss. Janney, Elstner Associates, Inc., Emeryville, CA 94608

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INTRODUCTION

Design or Analysis

Inelastic seismic design methodologies generally include an analytical procedure that is required to verify the design of the building. The question often asked is whether it is design or analysis. The answer is both. Analysis is part of any design procedure. For example, sizes and shapes are selected, forces and deformations are calculated, the sizes and shapes are verified by analysis, they are revised and verified again if required, and final details are designed.

Use of the Procedures

The procedures are used for both new and existing buildings. For new buildings they can be used to verify whether a code-based design (e.g., UBC) satisfies performance goals (e.g., damage control, life safety) for earthquakes with specified return periods. For existing buildings they can be used to evaluate performance characteristics of the as-is condition or to evaluate various levels of upgrading for performance goals and earthquake intensities. The procedures can also be used for post-earthquake evaluations to compare analyses with observations and to quantify damage and loss of capacity.

Inelastic vs Elastic

Although an elastic analysis gives a good indication of the elastic capacity of the overall structure and indicates where first yielding will occur, it cannot predict failure mechanisms and account for redistribution of forces during progressive yielding. Inelastic analysis procedures help under~d how the building really works by identifying modes of failure and the potential for progressive collapse.

Idealized examples of inelastic behavior patterns of a building with a moment resisting framing system are given in Figure 1. When lateral forces are applied to the frame an elastic distribution of forces to the components can be ~alculated. For example. the axial forces at the end columns will be greater than the forces at the interior columns (diagram a). If the center-bay beams are the first to yield, and if the progression of yielding remains in the center bay as the lateral displacements increase, the framing system will start to act as two individual end bays with yielding link beams in the center (diagram b). In this case, the axial loads in the columns will redistribute and the interior columns will reverse from tension and compression to compression and tension, respectively. A different scenario occurs if the exterior beams are the first to yield (diagram c). In this case, the center bay will pick-up the redistributed loading and there will be a substantial increase of axial forces on the interior columns. A third possibility is that of a weak first story where yielding of the lower column leads to the failure mode.

Generalized Inelastic Procedures

The various procedures can be placed into five generalized categories as follows: ( 1) Inelastic demand ratio (IDR) method that uses elastic analysis procedures: (2) Displacement method that uses an equal displacement rule and pushover analyses; (3) Substitute structure

procedure that uses secant stiffness methods and substitute properties of elements; ( 4) Capacity Spectrum Method (CSM) that uses graphical procedures with pushover analysis, response spectra, and dynamic properties; and. (5) Nonlinear time-history procedures (not in scope of this paper).

Terminology

Some of the new terminology resulting from the development of these procedures includes: "equal displacement rule", "inelastic response spectra", "pushover analysis", "secant modulus method", "substitute structure", "capacity spectrum method", "surrogate" or "equivalent" damping, and "multimode pushover". There needs to be a uniform explanation of the meaning of these terms and how they are used.

Developing Methodologies

Performance based seismic provisions are being developed for the following projects: FEMA project ATC-33 (ATC 1995A), State of California Proposition 122 project ATC-40 (ATC 1995B), SEAOC project Vision 2000 (SEAOC 1995), Department of the Army contract to update TriServices manual TM 5-809-1 0-l on dynamic analysis procedures (Army 1995), and the City of Los Angeles proposal for upgrading masonry infilled concrete frame buildings (Los Angeles 1995). Currently, ATC-33 appears to be heading for variations of the IDR method and the displacement method. ATC-40 is incorporating the CSM with guidelines for simplified methods. Vision 2000 appears to be giving a look at each of the developing methodologies. The · TriServices manual is updating the IDR and CSM procedures. The City of Los Angeles is proposing a secant methodology.

INELASTIC DEMAND RATIOS (IDR)

The IDR method was first introduced in TriServices technical manual TM 5-809-10-1 (Army 1986) and is presently being updated in a revised version of the manual (Army 1995 draft). An elastic modal (dynamic) analysis is made using the design earthquake response spectrum without the use of Rw or R. Reduced stiffnesses are used in the mathematical model to allow for some lengthening of the periods of vibration due to high stress demands and inelastic response (e.g., the elastic period may be increased by a factor of 1.4). A higher damping value is allowed to account for the effects of high stress levels and yielding of structural components. (e.g., damping increases from 5 percent to 10 percent). For the purists, this need not be interpreted as a viscous damping, but should be considered as a surrogate. For each component of the structure, the ratios of the calculated demand to the capacity of each component (i.e., DCR, demand capacity ratio) are compared to allowable consensus values of IDR's. A review of the distribution of OCR's is made in an attempt to identify unusual progressions of failure. Because this is an elastic procedure, the allowable IDR's are generally considered to be on the conservative side.

Figure 2 illustrates the force reduction effects of the longer period and increased damping for two hypothetical cases. Case A represents a short period building where the forces are at the plateau of the earthquake demand curve. Figure 2, which is shown in units of spectral

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acceleration (S~ vs spectral displacement (Sd) can also be expressed in terms of base shear (V) ·vs roof displacement (fl.J. The period is represented by the sloping lines. In Case A the elastic demand is point Al. The lengthening of the period increases the displacement and the increase in damping reduces the forces. The inelastic demand is represented by point A2. The longer period building is represented by Case B. In thiscase. the elastic demand is point Bl. In this case the forces (accelerations) are reduced by both the lengthening of the period and the increased damping (point B2). Although the displacement is increased by the period lengthening, a substantial amount of the increase is offset by the reductioh due to increased damping.

PUSHOVER ANALYSIS

The maximum force that a structure can be subjected to is dependent solely on its strength. In other words, it cannot resist mo.re force than a structure is capable of resisting (i.e., it just keeps displacing). This can be demonstrated by the pushover analysis. The idea of the pushover has been around for some time (Blume. et al. 1961 ). Lateral forces are applied to a structure until some components reach their elastic limit or yield point. Components that are about to yield or develop plastic hinges are relaxed (or softened) to allow them to yield without taking much (if any) additional force. As additional lateral forces are applied to the mathematical model. the loads are redistributed to the remaining elastic components until additional components reach their elastic limit. This procedure is repeated until the lateral-force-resisting system can no longer take additional force (i.e., a plastic mechanism forms, excessive displacement occurs that causes vertical instability, components degrade. or brittle failure occurs). This limit state at which the structure can resist no further increase in load will be called the ultimate capacity and will be designated by V u and fl. u• where V u is the base shear and fl. u is the displacement of the roof. A sample pushover curve is shown in Figure 3. Point A represents the point of first yielding with a base shear (V) of 2200 kips and roof displacement (~R) of 2.3 inches. The weight (W) of the building is 10,000 kips and the elastic period is 0.80 seconds. After relaxing the initial yielding components, the structure is pushed to point B. where additional components are relaxed due to yielding. The newly softened model is pushed from points B to C. softened some more. and then pushed to point D where mechanisms are forming and the cumulative base shear has increased to 3000 kips (V u) and the roof displacement is at 8.7 inches (~u>·

It should be noted that the pushover represents the global action of the whole building. For example, at each increment of base shear (V), the story forces are applied in proportion to

the product of story weight times story mode shape factor (i.e., Fx = [wx<f>x/:Ewx<f>x]V). Cumulative V is plotted against cumulative roof displacement ~Rat each increment. Up until first yielding everything is elastic. both globally (whole building) and locally (each component of the structure). A little beyond the first yielding, there is some minor local yielding of a limited number of components, but the overall structure remains essentially elastic. As the building goes into a recoverable inelastic state, globally the structure exhibits minor inelastic tendencies, while there is major local yielding of some components, minor yielding of others, but a great many components remain elastic. During major global inelastic behavior many localized plastic hinges may form; there will be minor local yielding of some components and there will still be many components that remain fully elastic. Figures 4 and 5 illustrate the difference

between local and global yielding. Whereas a point along the global capacity curve represents a system ductility; at that point. ductility demands on individual components (i.e., localized component ductility) can range from zero to very large. The pushover curve generally represents the inelastic force-displacement curve of the fundamental mode of the structure. Pushover curves may also be constructed for higher modes of vibration (e.g., second. third. etc.) as discussed later in this paper.

RESPONSE SPECTRA

ADRS Format

Response spectra have been traditionally plotted on a tripartite log format (i.e., three-way log paper) or a linear spectral acceleration (SJ vs period (T) format. An acceleration­displacement response spectnnn (ADRS) format has been introduced {Mahaney et al., 1993) that plots Sa vs Sd (spectral displacement) with T represented by lines radiating from the origin. This format is useful for two reasons: ( 1) both acceleration and displacement can be visualized on the same plot and (2) a capacity curve can also be overlayed on the same plot as will be discussed in the section on the CSM. Constant velocity lines can also be included on the format as curved lines (Figure 6). ~//

/X-) Inelastic Response Spectra

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Response spectra are generally plotted on the basis of elastic response of single lumped mass. single degree of freedom oscillators. Research studies have been done on developing inelastic response spectra that represent the nonlinear response of single mass oscillators for given ductilities (Newmark and Hall 1982; Miranda and Bertero 1994: Seneviratna and Krawinkler 1994 ). The results of these studies indicate that for a standard response· spectrum, with constant acceleration, constant velocity, and constant displacement zones. the inelastic response when compared to the elastic response. is as follows: ( 1) accelerations are smaller and displacements are greater in the constant acceleration zone: (2) accelerations have larger reductions. but the displacements are approximately equal in the constant velocity zone: and (3) both acceleration and displacement are smaller in the constant displacement zone. These relationships are trends based on statistical evaluations of limited data. There is a substantial deviation from the mean for individual points of data, thus caution should be used when attempting to accurately predict the response of individual structures to specific earthquake ground motion.

Surrogate Damping

The substitution of a higher value of damping is sometimes used to approximate the inelastic response of a structure. Academics often object to the use of "viscous" damping to approximate inelastic response: however. methods for developing equivalent viscous damping ratios have been available for some time (Jacobsen 1959, URS/Blwne 1975). A recent study by the author compared Newmark and Hall procedures for approximating inelastic response spectra as a function of ductility ratios to their procedure for calculating response spectra for various damping values. There is a correlation between the relative values of ductility (p.) and damping

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ratios (p). Figure 7 illustrates this relationship for system ductilities (J.ls) and surrogate damping ratios {Ps) on the basis of an elastic response spectrum with 5 percent damping for a one sigma cumulative probability. For example, a 15 percent damped response spectrum will represent an inelastic response spectrum for a system ductility demand of J.15=2.0,. The reduction in the constant acceleration range is 58 percent of the elastic spectrum and in the velocity range (curved portion); it is at 70 percent These values are slightly different for a median probability spectrum. The use of the surrogate damping spectrum will be discussed in the section on the CSM.

DISPLACEMENT METHOD

Elastic vs Inelastic Displacement

The displacement method is based on the assumption that the inelastic displacement of a single-degree-of-freedom system can be approximated from the elastic displacement that the structure would experience if the building were to remain elastic (Moehle 1992). The method has been extended to multi-mass, multi-degree-of-freedom systems. The relationships between elastic and inelastic are based on studies described in the discuSsion on inelastic response spectra.

Equal Displacement Rule

Research studies have lead to a displacement procedure developed on the basis of the · equal displacement rule (Moehle 1992; Rutherford & Chekene 1994) that applies when the elastic response falls within the constant velocitY portion of the 5% damped response spectrum (i.e .. the curved portion). Intuitively, it is difficult to understand why the equal displacement rule is valid. In other words. how does the structure know what it would do if it remained elastic in order to displace an equal amount inelastically? However, if all the structures have similar capacity characteristics and the demand spectra have similar characteristics. then it makes sense that the results would follow a pattern and that the structures could coincidentally have an inelastic displacement equal to the elastic displacement. For a structure with a short period. such as in the constant acceleration portion of the response spectrum, a displacement amplification factor is used to approximate inelastic displacements, which are greater than elastic displacements (Senevirama and Krawinkler 1994; Moehle 1992; A TC 1995A).

Description of the Procedure

The structure is mathematically modeled in accordance with a prescribed set of rules (i.e .. cracked concrete sections, etc.) and analyzed to determine the lateral displacement of the building for the design earthquake on the basis that the structure remains elastic. This is called the target displacement for long period buildings (curved portion of response spectrum). This displacement is multiplied by a factor (i.e., C0 ) to obtain the target displacement for short period buildings. The structure is then subjected to a pushover analysis to determine if the structure can be displaced to the target displacement without exceeding the inelastic capacities of the structural components. The procedure is illustrat~d in Figure 8 on an ADRS format. with Building B representing the long period building and Building A representing the short period building. The target displacements are A 1•

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SECANT METHOD

Substitute Structure

The following procedure, which is the author's interpretation of a methodology developed by Mete Sozen and others (Shibata and Sozen 1976), leads to a secant method of design. It is assumed that the elastic structure can be modified using a substitute stiffness and substitute damping to form a substitute structural model that can be analyzed elastically. Judgement or prescribed guidelines are used to assign secant stiffnesses to each component of the structural system (e.g., KE through K4 in Figure 5) to represent assumed member ductility demands. A substitute (i.e., surrogate or equivalent) damping value is assigned to represent the ductility demand on the overall structure (e.g., Figure 4).

Elastic Analysis

The mathematical model of the substitute structure is analyzed for the demands of the substitute response spectrum (e.g., modal analysis using the higher damped response spectrum). A graphical example is shown in Figure 9. For Building A. the short period case, the substituted structure has a period of 0.65 seconds with Sa equal to 0.58g and D., equal to 2.4 inches. The elastic structure, represented by the dashed line has a period of 0.4 seconds. For Building B, T equals 1.4 seconds, ~a equals 0.3g and Sd equals 5.7 inches. The elastic period is just short of 1 second.

Verification of Results

The results of the elastic analysis must be reviewed to determine if the demands on each of the structural components are consistent with the assumed secant stiffnesses. If the demands are too high, the secant stiffness should be reduced. If the demands are small. the secant stiffness should be increased or changed back to the elastic stiffnesses. The fundamental period of vibration (secant period. Figure 4) must be reviewed. relative to the elastic period (T E) to verify that the system ductility demand is consistent with the assumed substitute damping. If revisions are required. the analysis is repeated with the revised mathematical m~

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CAP A CITY SPECTRUM METHOD

Methodology

The Capacity Spectrum Method (CSM) was originally developed as a rapid evaluation method for a pilot seismic risk project of the Puget Sound Naval Shipyard for the U.S. Navy (Freeman. et al. 1975). It was later used as a procedure to correlate earthquake ground motion with observed· building performance (Freeman 1978 and A TC 1982) and has been incorporated in the TriServices Seismic Design Guidelines for Essential Buildings (Army 1986) as part of the two-level approach to seismic design. The methodology has also been included in other documents for evaluation and upgrade procedures (ATC 1991, NIST 1994). The CSM has the

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feature of rationally explaining how buildings respond to earthquake ground motion. It takes a graphical representation of the global force-displacement capacity curve of the structure (i.e., pushover) and compares it to the response spectrum representation of the earthquake demands.

The capacity curve is determined by statically loading the structure with lateral forces to calculate the roof displacement and base shear coefficient that defines first significant yielding of structural elements. The yielding elements are then relaxed to form plastic hinges and incremental lateral loading is applied until a nonlinear static capacity curve is created. The curve is created by superposition of each increment of displacement and includes tracking displacements at each story (A TC 1982). This procedure is the pushover analysis discussed above. It is assumed that the structure can take a number of cycles along the capacity curve and behave in a hysteretic manner. The stiffness is assumed to reduce to an equivalent global secant modulus measured to the maximum excursion along the capacity curve for each cycle of motion. The roof displacement and base shear coefficient coordinates are converted to spectral displacements (Sd) and spectral accelerations (Sa), respectively, by use of modal participation factors and effective modal weights as determined from dynamic characteristics of the fundamental mode of the structure. Note that the dynamic characteristics change as the building response becomes inelastic. .-ill equivalent inelastic period of vibration (Ti) at various points along the capacity

curve is calculated by use of the secant modulus (i.e .• T, = 21tJSd; IS"' g) . Now the capacity

spectrum curve can be plotted on an ADRS format.

The demand curve is represented using several earthquake response spectra with various levels of damping. For example, the 5 percent damped curve may be used to represent the demand when the structure is responding elastically. The 10 percent and 20 percent damped curves may be used as a surrogate to represent the reduced demand in the inelastic range to account for hysteretic damping and anti-resonant nonlinear effects. The surrogate damped curves are used as a tool for reducing the spectrum to an equivalent inelastic spectrum.

. Both the capacity and demand curves are plotted together on the ADRS format. The Capacity Spectrum Method can be summarized as follows:

If the capacity curve can extend through the envelope of the demand curve, the building survives the eanhquake. The intersection of the capacity and appropriate surrogate damped demand curve represents the inelastic force and displacement of the structure.

Graphical Solution

Figure 10 is an idealized example of the CSM that is comparable to the examples for the displacement method (Figure 8) and the secant method (Figure 9). The solid response curve represents the elastic demand spectrum and the dashed response curve represents the surrogate inelastic response spectrum. The short period Building A and the longer period Building B are represented by the two capacity spectra. The intersections with the surrogate spectrum at .1 1, represent the inelastic spectral displacements of the structures as well as the inelastic spectral accelerations.

Conversion from v vs AR (Capacity Curve) to sa vs sa (Capacity Spectrum)

As stated above, modal participation factors and effective modal weights are used to convert v vs AR to sa vs sd. For those not familiar with the process, the. following example is given (a more detailed example may be found in references Army 1986, A TC, 1982, and Army 1995).

In the pushover curve of Figure 3, point A has the coordinates of V=2200 kips and AR=2.3 inches. The weight of the building is 10,000 kips, so the base shear coefficient C8 is equal to V!W=0.22. The fundamental modal participation factor, the ratio of roof displacement to the spectral displacement, is calculated to be 1.30 (i.e., AJJSd=l.30) by dynamic analysis. Therefore. ScrAJ1.30=2.3/1.30=1.77 inches. The fundamental effective modal weight ratio, the ratio of C8 to Sa, is calculated to be 0.78 (i.e., C8/Sa=0.78) by dynamic analysis. Therefore, Sa=Csf0.78=0.22J0.78=0.28g. Thus, the capacity spectrum coordinates for point A are Sa=0.28g and Sd=l.77 inches.

Use of CSM with Recorded Earthquake Motion

Figure 11 shows a capacity spectrum for each of two steel frame buildings (Freeman, 1987). Figure 11A shows response spectrum for UBC site S2. Figures 118, C, and D show 5% damped response spectra obtained for the 1994 Northridge earthquake at Santa Monica, Arleta and Holiday Inn at Van Nuys, respectively. Figure liD also shows the N-S response spectrum for the 1971 San Fernando earthquake. By plotting the capacity spectrum of Figure 11 on Figures llA, B, C, and D, the variations in response to the different earthquakes are apparent.

Multi-Mode Analysis

The examples shown are valid assuming the response is governed by the fundamental mode of vibration. However, for some buildings, espe<;ially tall long period buildings, the higher mode effects may be critical. One method of including higher mode effects is described in the TriServices manuals (Army 1986, 1995). Another method being developed is shown in Figure 12 .. In this procedure pushover analyses are done for additional mode shapes. For example, force distributions are applied to deform the building into the second and third translational mode shapes. Yield patterns will be substantially different than those obtained for the first mode shape. The V vs AR values for the higher modes are converted to Sa vs Sd curves using the higher mode participation factors and effective modal weights. These curves are plotted on the ADRS format and the demands on each of the modes can be determined. Earthquakes with response spectra of varying shapes (e.g., Figures liB, C, and D) will have significantly different effects on the structure. Each component of the structure is then evaluated for the different modes for ductility demands.

CONCLUSIONS

Although the proposed methodologies appear to approach the problem of inelastic response from different points of view, they all have some elements in common and are generally based on the fundamentals of statics, dynamics, and the mechanics of materials. In addition, the methodologies are aided by experimental data, statistics, and judgmen~. If all the methods are valid, they should give similar results. Comparisons of various methodologies, with their attributes and weaknesses, leads to a better understanding of inelastic response of structures and improved seismic design and evaluation procedures.

All the methodologies tend to give similar results under ideal conditions such as shown in the examples in Figures 8, 9, and 10. The IDR values are established to be on the conservative side, as shown in Figure 2, because the method does not investigate inelastic redistribution. The results of all the methodologies become more divergent when structural systems are atypical. response spectra shapes are irregular. and material properties are unusual. Although the author may be biased. it appears that the CSM is better able to account for atypical conditions.

Because of time and space limitations, all aspects of this subject cannot be addressed and the reader is encouraged to explore the reference material. Your comments on this paper are welcome.

REFERENCES

Army, 1986, Seismic Design Guidelines for Essential Buildings, Departments of the Army (TM 5-809-10-1), Navy (NAVFAC P355.1), and the Air Force (AFM 88-3, Chap. 13, Section A, Washington, D.C.

Army, 1995 DRAFT (unpublished), Seismic Dynamic Analysis for Buildings (Revision to Army TM5-8-9-1 0-1 ).

ATC, 1982, An Investigation of the Correlation between Earthquake Ground Motion and Building Performance (ATC-10), Applied Technology Council, Redwood City, California.

ATC, 1991, U.S. Postal Service Procedures for Seismic Evaluation of Existing Buildings (Interim), Applied Technology Council, ATC-26-1 Redwood City, California

ATC, 1995A DRAFT, Guidelines and Commentary for the Seismic Rehabilitation of Buildings; (2 volumes), Draft (75% Submittal Third Draft), developed under contract to Building Seismic Safety Council, ATC-33.03, Applied Technology Council, Redwood City, California

ATC, 1995B Rough DRAFT, Product 1.2 Recommended Methodology for Seismic Evaluation and Retrofit of Existing Concrete Buildings , for the Seismic Safety Commission, Sacramento, California (unpublished), ATC-40, Applied Teclmology Council, Redwood City, California.

Blume, John A., N.M. Newmark, and Leo M. Corning, 1961, Design of Multi-Story Reinforced Concrete Buildings for Earthquake Motions, Portland Cement Association, Chicago.

Freeman, S.A., J.P. Nicoletti, and J.V. Tyrell, 1975, Evaluations of Existing Buildings for Seismic Risk - A Case Study of Puget Sound Naval Shipyard, Bremerton, Washington. Proceedings of the U.S. National Conference on Earthquake Engineers. EERI. pp 113-122, Berkeley.

Freeman, S.A., 1978, Prediction of Response of Concrete Buildings to Severe Earthquake Motion, Douglas McHenry International Symposium on Concrete and Concrete Structures, SP-55, pp 589-605, American Concrete Institute, Detroit.

Freeman, S.A., 1987, Code Designed Steel Frame Performance Characteristics, Dynamics of Structures Proceedings, pp 383-396, Structures Congress '87, Structural Division, American Society of Civil Engineers, Orlando. ·

Jacobsen, L.S., 1959, Frictional Effects in Composite Structures Subjected to Earthquake Vibrations, Stanford University.

Los Angeles, 1995 DRAFT, Division XX, Earthquake Hazard Reduction in Existing Reinforced Concrete Buildings and Concrete Frame Buildings with Masonry lnfills, Draft 1/31/95 (unpublished).

Mahaney, J.A., Terrence F. Paret, Brian E. Kehoe, Sigmund A. Freeman, 1993, The Capacity Spectrum Method for Evaluating Structural Response during the Lorna Prieta Earthquake, 1993 National Earthquake Conference, Memphis.

Miranda, E. and V.V. Bertero, 1994, Evaluation of Strength Reduction Factors for Earthquake­Resistance Design, Earthquake Spectra, EERI, Vol 10, No 2, Oakland, California

Moehle, J.P., 1992, Displacement-Based Design of RC Structures subjected to Earthquakes, Earthquake Spectra, Vol 8, No 3, EERI, Oakland, California

Newmark, N.M., and W.J. Hall, 1982, Earthquake Spectra and Design, Earthquake Engineering Research Institute.

NIST, 1994, Standards of Seismic Safety for Existing Federally Owned or Leased Buildings, U.S. Department of Commerce, National Institute of Standards and Technology, NISTIR 5382, ICSSC RP-4, Gaithersburg, Maryland.

Rutherford & Chekene, 1994, Provisional Commentary for Seismic Retrofit, Product 1.1 of the Proposition 122 Seismic Retrofit Practices Improvement Program, California Seismic Safety Commission, Report No. SSC 9402.

SEAOC, 1995, Vision 2000 Progress Report, Proceedings of 1995 Convention.

Seneviratna, G.D., and H. Krawinkler, 1994, Strength and Displacement Demands for Seismic Design of Structural Walls, Proceedings of the Fifth U.S. National Conference on Earthquake Engineering, Chicago.

Shibata, A. and M.A. Sozen, 1976, Substitute-Structure Method for Seismic Design in fljC, Journal of the Structural Division, ASCE, Vol 102, No 1, 1-18.

URS/John A. Blume & Associates, Engineers, 1975, Effects of Prediction Guidelines for StrUctures Subjected to Ground Motion", Roger E. Scholl. Compiling Editor, JAB-99-115, UC-11, San Francisco.

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..f-' ()

, ~0.2 (f)

0 0

T=1.5

~~~~~*~~-·~~~~T=J.O T=ti.O

I I

2 4 6 8 10 12 Spectral Displacement, Sd (inches)

FIGURE 12. Multi-mode capacity spectra