1 European School for Advanced Studies in Reduction of Seismic Risk Research Report No. ROSE-/ Dynamic Behaviour of Reinforced Concrete Frames ed with Direct Displacement-Based by J. Didier Pettinga Graduate Student M.J. Nigel Priestley Co-Director ROSE School c/o EUCENTRE Via Ferrata,, Pavia, Italy May
3 TABLE OF CONTENTS Table of contents...v. INTRODUCTION.... OBJECTIVES AND INTENT.... CURRENT DESIGN APPROACHES.... INTRODUCTION.... FORCE-BASE DESIGN METHODS..... Equivalent Static Lateral Force Method..... Multi-modal Superposition.... DISPLACEMENT-BASED DESIGN METHOD.... DIRECT DISPLACEMENT-BASED DESIGN..... Introduction..... Direct Displacement-Based Method.... DDBD FRAME DESIGN PROCEDURES.... GENERALISED DESIGN PROCEDURE.... DESIGN PROCEDURE AS APPLIED TO THIS STUDY..... Frame Descriptions..... Displacement Spectrum..... Direct Displacement-based Parameters..... Section Analysis.... INELASTIC TIME-HISTORY ANALYSES.... MODELLING ASSUMPTIONS.... SPECIFIC MODEL DETAILS AND DEFINITIONS..... Model type and geometry..... Member bilinear factors and plastic hinge properties..... Modified Takeda Hysteresis rule..... Viscous damping..... P- Effects..... Input ground motions.... INITIAL INELASTIC TIME-HISTORY RESULTS.... MAXIMUM DRIFT AND DISPLACEMENT PROFILES.... OBSERVATIONS AND DDBD METHOD CHANGES FOR FRAMES...
4 J. Didier Pettinga & M.J. Nigel Priestley.. displacement profile developments..... Distribution of beam strengths..... Dynamic amplification of storey drifts.... SUMMARY OF PROPOSED CHANGES TO THE DIRECT DISPLACEMENT- BASED FRAME DESIGN PROCEDURE..... IMPLEMENTATION OF SUGGESTED DDBD PROCEDURES.... REVISED DIRECT DISPLACEMENT-BASED DESIGNS.... INELASTIC TIME-HISTORY RESULTS USING REVISED DDBD METHODS..... Description of results..... Comments on results.... ACCOUNTING FOR DYNAMIC AMPLIFICATION.... APPLICATION OF EXISTING METHODS..... FORCE-BASED AMPLIFICATION APPROACHES..... DDBD DYNAMIC AMPLIFICATION METHOD: MODIFIED MODAL SUPERPOSITION.... DEVELOPMENT OF DYNAMIC AMPLIFICATION PROCEDURES FOR. FRAME DESIGN..... Column shear force amplification..... Column bending moment amplification... VERIFICATION WITH REAL EARTHQUAKE ACCELEROGRAMS.... MODIFICATION FOR TWO-WAY FRAMES.... SUMMARY OF PROPOSED METHODS OF ACCOUNTING FOR DYNAMIC AMPLIFICATION.... PARAMETRIC INVESTIGATION AND FURTHER DISCUSSION.... VARIATION OF BEAM DEPTH..... Time-history results and application of amplification methods.... INCREASED BEAM SPANS.... DRIFT AMPLIFICATION DEPENDENCY ON INTENSITY.... SUMMARY OF PARAMETRIC STUDY.... CONCLUSIONS.... DDBD PROCESS: REVIEWS AND DEVELOPMENTS...
5 Dynamic Behaviour of RC Frames designed with Direct Displacement-Based. DDBD METHODS OF ACCOUNTING FOR DYNAMIC AMPLIFICATION IN FRAMES.... FUTURE DEVELOPMENTS... REFERENCES... APPENDIX A TIME HISTORY RESULTS USING REAL ACCELEROGRAMS
7 . INTRODUCTION Reinforced concrete frame structures are a common building form in seismically active regions. However, the use of such structural forms and the inherent variability in geometric proportions, sectional shapes and material properties means that the dynamic behaviour under earthquake loading has been, and remains, difficult to consistently evaluate. Developments in seismic design of reinforced concrete structures can largely be attributed to research aimed at better understanding the mechanics of concrete behaviour and thus improving the detail design methods applied to structural and non structural elements. This has reached the extent that in some forms, code requirements are such that newly designed buildings can be considered sufficiently safe under seismic excitation. Fundamental to the process of seismic design, has been the use of force-based design whereby a set of external forces, seen as equivalent to the inertial forces induced by ground accelerations, are applied to the structure. These design methods are often based on the key assumption that dynamically, the structure will behave principally with first mode response, with requirements for using multi-modal analysis when certain limits structural behaviour are not met. In many situations the first mode dominance is a valid assumption, however it is not necessarily satisfactory to disregard the presence of vibration modes above the fundamental, the so called higher modes. Studies reported by Paulay (), and further presented by Paulay and Priestley (), showed that higher modes do in fact play a significant role in reinforced concrete frame and wall behaviour, and that approaches to encompass these effects must be considered, especially when using simplified analysis methods such as the equivalent static lateral force-based approach. Current code provisions in New Zealand (NZS ; and NZS ;) have adopted the procedures described by Paulay and Priestley (), thereby accounting for the presence of higher mode effects using simplified equations that act to scale-up the design moments and shears such that the design envelope should provide sufficient capacity to prevent exceedance under earthquake demands.
8 J. Didier Pettinga & M.J. Nigel Priestley Recent work presented by Priestley and Amaris () looked at the dynamic behaviour of reinforced concrete structural walls designed using direct displacement-based design. The results added further emphasis to the importance of dynamic amplification, and emphasises that the current code provisions are in many cases insufficient. As a result of this work a modified form of the commonly used elastic modal superposition, was put forward as a simple and generally very effective approach to accounting for dynamic amplification of wall shear forces and bending moments. Preliminary studies on frame behaviour designed with direct displacement-based design by Priestley () have however suggested that the method found for wall structures does not transfer to frames in a succinct manner. Results were generally inconsistent and often gave a poor fit to mean envelopes found from inelastic time-history analyses. Hence the need for further work to develop a rational and simple approach for frame structures.. OBJECTIVES AND INTENT Because the equations applied in design codes for frame dynamic behaviour were developed using force-based design procedures, it is important to check these methods with displacement-based design approaches to verify if and what differences there are in the dynamic behaviour of buildings designed with such processes. The apparent inability of the modified modal superposition approach to accurately predict the dynamic behaviour of frame structures is due to many factors. However, one particular issue is that of the variability in geometric parameters. To this extent the research results presented here, attempts to narrow the geometric bounds, such that the behaviour of one particular form of simple frame structure can be assessed. The use of relatively deep beams and short spans has been adopted in what might be termed a tubeframe structure, where by the perimeter frames are intended to act as the principal lateral deformation resisting components. This definition has also been made to ensure reasonable ductility demands by generating relatively stiff structures with low yield displacements. Using code drift limits normal reinforced concrete -way frames have ductility demands of about two, hence they would not provide a rigorous investigation of ductile dynamic behaviour. The intent of this work is to investigate some key aspects of the direct displacementbased design method, using designs for,,,, and storey frames. The following points define the objectives and general method. Using design spectrum compatible accelerograms, a series of inelastic time-history analyses of varying intensity are used to help assess the demands resulting from dynamic amplification.
9 Dynamic Behaviour of RC Frames designed with Direct Displacement-Based A systematic investigation of the design displacement profiles and their application to ensure that drift demands satisfy the specified design drift limits. This is particularly important with respect to the ability of the method to control higher mode drift effects. An examination of the significance of higher mode contributions to column shear forces and bending moments, in particular with respect to ductility demands and earthquake intensity.
11 . CURRENT DESIGN APPROACHES. INTRODUCTION The standard approach for seismic design of structures, specified in many design codes, involves the development of a base shear value corresponding to a given design spectrum. This design value is often derived for a reduced level of excitation on the proviso that ductile inelastic behaviour is acceptable and allowed for at section and member levels by providing adequate detailing to ensure material behaviour that accommodates the strain demands. From the reduced design spectrum a set of lateral force vectors can be developed to represent first mode response only, as is the case for equivalent static methods, or as a series of force vectors representing a number of modes as specified in code requirements. These lateral forces are then applied to the structure as external forces and the member actions resulting, taken directly for design or statistically combined to give final design values. Thus the underlying intent of force-based design involves the specification and attainment of a minimum strength, based on assumptions of initial stiffness, design earthquake intensity and ductile capacity of the structure, as applied through a force reduction factor. In some cases it is required to check the structural deformations, often in the form of storey drifts, against code defined limitations, to verify the deformation behaviour of the structure. This approach contrasts strongly to that of direct displacement-based design whereby structures are designed to attain a specified level of damage commensurate with a given limit state. By specifying a target displacement-based on an equivalent single degree of freedom system, it is possible to derive values of strength, stiffness and thus structural period. This method is further outlined in Chapters and.. FORCE-BASED DESIGN METHODS The following sections outline the methods of generating design forces described above, including code recommendations for the application of the methods.
12 J. Didier Pettinga & M.J. Nigel Priestley.. Equivalent Static Lateral Force Method This approach utilises the key assumption that the first mode dominates the structural response to the extent that a direct account of higher modes is not required. Code limitations on the use of this method generally centre around having a first mode period less than a specified maximum and maintaining the vertical and horizontal regularity of the structure. If a structure satisfies these requirements then the force vector is defined such that the distribution varies linearly up the structure as an inverted triangle. This profile thus matches the fundamental mode shape with no direct account for the higher modes. Some provision can be made, through the use of an additional proportion of force at the roof level. This extra force is specified as % in NZS, % by Paulay and Priestley (), and ranges from to % for the UBC. It is noted that Eurocode (EC) does not specify this additional force. The fundamental period of the structure can be estimated through the use of established equations such as that of the Rayleigh Method (NZS :, EC, UBC): N ( Wi i ) T = π (.) g N ( Fi i ) or with simple correlated approximations as suggested in EC: T Ct H = (.) In which C t =. for moment resisting concrete frames up to m in height and H being the height in meters. Using design spectra, a value of Spectral Acceleration S a (T ) is obtained from which the base shear can be calculated using the total seismic weight of the structure W t. V b a ( T ) Wt = S (.) The base shear can then be distributed as an equivalent lateral force vector at each level i using a generalised form:
13 Dynamic Behaviour of RC Frames designed with Direct Displacement-Based F = V i b N ( Wihi ) i= W h i i This form is altered in NZS to give the additional proportion of base shear applied at the roof level, intended to account for increases in shear and moment values due to higher modes: F = F i t +. V b N ( Wihi ) i= where Ft =. V b at roof level and for all other levels. The lateral force distribution resulting from this equation is seen in Figure -. With this distribution of forces the global overturning and storey shear forces can be found along with individual member actions. W h... Methods of Dynamic Amplification. It is clear that for taller structures this simplified approach is not adequate to account for modes above the fundamental, which tend to take greater participation as overall height increases. Thus it is foreseeable that further account could be made at a later stage of the design process. This has been used in NZS following the methods presented by Paulay and Priestley () for frames and walls. Moment Amplification: Equations are presented in NZS for both one-way and twoway frames, to account for concurrent moment capacity development in beams framing into a column, shifts in column point of contraflexure and in the case of two-way frames the reduced section efficiency for moments applied about the section diagonal axis. The amplification factor is given by ω and takes the following forms and limits for each frame type: i i (.) (.)
14 J. Didier Pettinga & M.J. Nigel Priestley + =.V b.v b.v b Figure -. The concept of the Equivalent Lateral Force method with additional roof level force One-way frames. ω =.T +.. (.a) Two-way frames. ω =.T +.. (.b) Here T is the fundamental period found by the methods above or some other means such as dynamic modelling with a computer program. As higher mode responses do not affect the required strength at column bases, the value of ω is taken as. for one-way frames and. for two-way frames to account for the reduced section efficiency about the diagonal section axis. These factors are also applicable at roof level where it is deemed acceptable for plastic hinges to form in the columns. With the intent of providing a gradual reduction in amplification, the minimum factor allowed in Eq.(.a & b) is used for the floor level immediately below the roof. Because higher mode response is seen to be significant in the upper storeys of a structure, Eq.(.a & b) is applied to the upper % of the building. From the base to.h, a linear variation in ω from. or. at the base, to the value calculated from Eq.(.a & b) is used. Thus the distribution of values for the dynamic amplification factor ω can be represented schematically as in Figure -.
15 Dynamic Behaviour of RC Frames designed with Direct Displacement-Based H M E. One-way Frame ω. Two-way Frame ω Figure -. Schematic representation of column bending moment dynamic amplification as suggested by Paulay and Priestley () for use with simplified lateral force methods The dynamic amplification factor is applied in the design moment equation as shown: M u = ωφ M (.) o where M E is the moment value from the equivalent lateral force analysis and φ ο is the flexural overstrength factor for plastic hinges that accounts for material overstrength and strain hardening effects. In design this factor is typically specified in relevant material design codes, such as NZS where a minimum value is defined as: E M φo = M o E λo. = = =. φ. This value of overstrength is often higher due to practical limitations on available reinforcing bar sizes leading to excess provisions of nominal strength. (.)
16 J. Didier Pettinga & M.J. Nigel Priestley For the purposes of this study the flexural reduction factor is taken equal to., thus M o = λo M E, with λ ο calculated and removed from the time-history results following the procedure defined in Section... Shear Amplification: It is possible to make an estimate of column shear by taking the amplified moment gradient found using the method above. It is likely that such an estimate could be excessively conservative. Therefore, with the probability of maximum amplified moments occurring simultaneously at each end of a column being sufficiently low, it is reasonable to consider another method of shear amplification. Paulay and Priestley () suggested a simple multiplicative factor that accounts for beam moment distribution into columns not following that given by an elastic analysis and thus generating a moment gradient, or shear force, larger than anticipated. This can be interpreted as a shift in the position of the point of contraflexure, and hence an account for dynamic magnification as suggested in the previous section. This approach has been adopted in NZS in the following forms: One-way frame upper storeys V u =.φ V (.a) o E Two-way frame upper storeys V u =.φ V (.b) With typical values of φ ο this can result in design values of approximately.v E and.v E for one-way frames and two-way frames respectively. In first storey columns where hinging is expected at the base, and should be anticipated at the top (due to inelastic first floor beam elongation), shear demands are calculated using the overstrength moments from top M o(top) and bottom M o(bottom) hinging: o E V col M M o( bottom) = (.) L o( top) + n where L n is the clear height of the column from the first floor soffit to the base. Note this should also be applied to the top storey, where it is possible that columns may form plastic hinges before the top floor beams... Multi-modal Superposition When a structure does not satisfy code requirements for any one of the following: maximum height, period, and horizontal or vertical regularity, a modal analysis and
17 Dynamic Behaviour of RC Frames designed with Direct Displacement-Based subsequent combination is generally required (NZS :, EC, UBC). This procedure requires the determination of a number of vibration periods in order to include a sufficient proportion of the structural mass in the analysis. EC specifies this proportion to be greater than %, with all modes contributing greater than % being included (NZS is similar). Because of the consideration of the modal contributions, this method is assumed to directly account for higher mode effects, especially in the upper levels of a structure, and hence no further amplification is specified for design purposes. Having determined these values of period and the corresponding mode shapes, the mass participation factors for each mode m can be found by the following: N φimmi i, m= ρ = (.) m N φim i, m= m i It is possible to obtain values of Spectral Acceleration, S a (T), from appropriate design spectra. From these values the base shear corresponding to each participating mode is found by: N i= m i V Bm = S ( ) a Tm g m N ρ m i (.) i= In a similar fashion to the equivalent static lateral forces described above, the base shear for each mode is distributed as: F im = V Bm N φ ( φimmi ) i, m= im m i (.) These forces are applied to the structure as external loads, thus giving the modal member actions. Maximum modal displacements are found from the pseudo displacement spectrum that can be directly calculated from the design spectral accelerations by:
18 J. Didier Pettinga & M.J. Nigel Priestley Tm m = S a ( Tm ) g (.) π This maximum displacement is used to find the modal displacement at each floor level i by the following: im N φ m im i m i, m= im N φimmi i, m= = φ (.) The results obtained from the above equations are specific to each vibration mode. Under excitation such as that due to earthquake ground motion, it is unlikely that the modal maxima will occur simultaneously. Thus it is considered that a direct summation of modal quantities will produce design values that are excessively conservative. For this reason it is common to apply one of two statistical combination methods; either the Square Root Sum of the Squares (SRSS) or Complete Quadratic Combination (CQC). The application of SRSS is limited to some extent by the requirement that two modes of vibration, i and j, are sufficiently separated in periods T i and T j, such that correlation between the two modes is not significant. EC specifies that SRSS may be used if this separation is such that T j.t i. The form of the SRSS combinations for equivalent lateral forces, shears and displacements are as follows: N F i = F im i, m= = N i, m= V i V im N i = i, m= (.) If the structure does not satisfy this constraint then a more rigorous combination that accounts for modal correlations using cross modal coefficients ρ ij can be used with the following form: F = F ρ F (.) m i j mi ij Because the modal superposition method uses values of spectral acceleration, it is common to reduce the design base shear and thus applied lateral forces to allow for ductile structural response. Assuming the equal displacement approximation is valid, this reduction can simply be taken as a direct division of the elastic forces (for all the modes used) by R (the force reduction factor) or q (the behaviour factor) equal to µ, the design mj im
19 Dynamic Behaviour of RC Frames designed with Direct Displacement-Based structural displacement ductility. This value is specified in design codes for various structural forms depending on the anticipated ductile capacity of the structure. For reinforced concrete frames the value of µ is usually taken between and.
21 . DISPLACEMENT-BASED DESIGN METHOD. DIRECT DISPLACEMENT-BASED DESIGN.. Introduction Recent developments in seismic design philosophy have centred around the concept of performance-based design. In the past years many researchers have proposed methods of design focusing on performance-based techniques (as summarised and compared by Sullivan et al., ). In its purest form this is an approach whereby a structure is designed, not to satisfy a set of performance requirements, but instead to meet the given criteria. In this way a structure is designed and expected to reach a specified acceptable amount of damage under one or more levels of design intensity earthquake. Structures designed in this manner can be expected to exhibit more of a uniform risk, something that is not easily achieved using force-based procedures. It should be noted that most performance-based methods check displacements at the end of the design process, or require transverse reinforcement detailing to match calculated rotations, few procedures proposed actually design to target drift amounts. Material strain limits are commonly used to define damage limit states. For reinforced concrete these can be defined by concrete compression strains and steel tension (or compression) strains. Often the serviceability limit state (SLS) is defined such that concrete compression strains remain lower than ε c =. above which point cover spalling may occur, and corresponding steel tension strains are low enough that residual crack widths are acceptably low, for example ε s =. to.. In a similar fashion, a damage control limit state can be defined based on larger strain levels. For concrete compression strains this can be a function of confined concrete strains that can be estimated from allowable transverse reinforcement tension strains caused by lateral concrete expansion, such that: (.ρ f ε ) s yh ε cm =. + (.) f ' cc suh
22 J. Didier Pettinga & M.J. Nigel Priestley Where ρ s is the confinement volumetric ratio, f yh the transverse steel yield strength, ε suh the transverse steel strain at maximum stress and f cc the confined concrete compression strength (Mander et al. ). Maximum longitudinal steel tensile strains can be set at any desired level, however for reasons relating to buckling (under reversed loading) and avoidance of low cycle fatigue (Priestley and Kowalsky, ) it is prudent to limit this level to the following: ε =. (.) sm ε su Further to the specification of material strain limits, it is possible, and conceptually more straightforward, to define limit states by drift amounts. For serviceability limits this could be set at θ =. to avoid non-structural damage, while for damage-control limit states, the amount of structural damage deemed acceptable, is left to choice, however, in many cases this drift limit will be specified in design codes. For instance NZS limits interstorey drifts to θ =. for designs completed without inelastic time-history analysis (for which θ =. is acceptable). Underlying the displacement-based design methods is the use of displacement design spectra. Often these are a simple conversion of spectral accelerations given in design codes. Of particular importance in the definition of a displacement spectrum is the so called corner-period beyond which peak displacements are assumed constant for increasing structural period. This reflects the fact that as structural period increases, maximum displacements tend to that of the peak ground displacement, thus for periods over a defined limit, peak displacements can be considered independent of structural period. The following section outlines one particular form of displacement-based design, namely Direct Displacement-Based (Priestley, ). The general method is described and appropriate equations defined, along with important assumptions underlying the design process... Direct Displacement-Based Method The following relations are a summary of the design process developed and described in greater detail by Priestley and Kowalsky () and Priestley et. al ().
23 Dynamic Behaviour of RC Frames designed with Direct Displacement-Based F m e F n F U rk i h e K i K E y d (a) (b) Elasto-Plastic. % Damping (%) Steel Frame Concrete Frame Unbonded Prestressing Displacement (m).... d % % % % T e Displacement Ductility Period (seconds) (c) Figure -. Fundamentals of Direct Displacement-Based [from Priestley et. al ()] The method of direct displacement-based design is founded on the use of a substitute structure as proposed by Shibata and Sozen () and shown in Figure -. This uses an equivalent single degree of freedom (SDOF) representation of a multi-degree of freedom (MDOF) system such that the natural period of the SDOF is given by: (d) T e me = π (.) K e where m e is the effective mass of the structure, K e is the effective stiffness at the design displacement and therefore: V = (.) b K e D with D representing the maximum design displacement of the substitute structure. To incorporate the effects of inelastic action in the real structure, hysteretic damping is combined with elastic viscous damping to give an equivalent viscous damping of the form:
24 J. Didier Pettinga & M.J. Nigel Priestley. ξ d = + ξ hyst % which for concrete beams becomes µ ξ d = + % (.) π and is thus applied to reinforced concrete frames as beam plastic hinging is the primary source of inelastic action. Note that µ is defined as the displacement ductility at the design displacement; µ = D y. Priestley () showed that the yield drift θ y can be calculated with sufficient accuracy for design purposes using the relation: l b y =. which multiplied by the H y e gives hb θ ε l b y =.ε y H e (.) hb With H e equal to the effective height. Thus the substitute structure yield displacement can be used to calculate the system displacement ductility, where l b is the beam length and h b the storey height. Note that this relation can also be applied to individual storey heights to allow for changes in design drift and beam depth up the height of the building. The displacement spectrum is used to directly evaluate the displacement demand of the equivalent substitute structure. However, for direct displacement-based design it may be necessary to adjust the displacement spectra for such use. Code based design spectra are normally developed for % damping (NZS, EC, UBC), hence to modify the spectra for damping levels ξ d other than %, the following equation from EC is used to give a corrected value of displacement demand for normal accelerograms recorded more than km from the fault rupture (the exponent β = ½): (, ) = (,) (.) T ξ T + ξ d Where ( T, ξ ) and ( T, ) are the design spectrum corner displacements at the calculated equivalent viscous damping level and % design level spectrum respectively. EC gives two values for the corner-period T D, depending on the earthquake type defined for a given region. For Type spectra, T D =. seconds, while for Type spectra T D =. seconds. However, recent work (Faccioli et al., ) has shown that the corner period is dependent on magnitude and epicentral distance, hence it would seem appropriate to extend the corner period beyond such low values as given by EC in order to develop results applicable to regions of higher seismicity. The equivalent substitute structure approximates the multi-degree of freedom structure at peak response, thus the effective stiffness of the structure is significantly lower than that for an elastic structure. Therefore the effective period T e is significantly longer and the
25 Dynamic Behaviour of RC Frames designed with Direct Displacement-Based displacement response spectra should be adjusted accordingly to account for this phenomenon. This is another reason for the extension of the corner period beyond that specified in many codes. Assuming a linear displacement spectrum (note this is not always the case) and an appropriate corner period with corresponding displacement demand, Eq.(.) and (.) can be substituted into Eq.(.) to give:,) ( + = = d D c c e D e b T m K V ξ π (.) with D found from: = = = n i i i n i i i D m m (.) where m i and i are the masses and design displacements at each significant level i of the structure up to n storeys. The effective mass m e is calculated in a similar fashion: D n i i i e m m = = (.) Further, the effective height of the substitute structure is found from: = = = n i i i n i i i i e m H m H (.) The final component of the direct displacement-based design process is an assumed displacement profile for the structure. As the equivalent substitute structure seeks to model peak response, the profile needs to reflect the inelastic deformed shape of the building. Earlier studies (Priestley, ; Loeding et al., ) proposed three separate
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Reading Assignment SLAB DESIGN Chapter 9 of Text and, Chapter 13 of ACI318-02 Introduction ACI318 Code provides two design procedures for slab systems: 13.6.1 Direct Design Method (DDM) For slab systems
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Performance-based Evaluation of the Seismic Response of Bridges with Foundations Designed to Uplift Marios Panagiotou Assistant Professor, University of California, Berkeley Acknowledgments Pacific Earthquake
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