A DIRECT METHOD FOR THE DESIGN OF VISCOUS DAMPERS TO BE INSERTED IN EXISTING BUILDINGS

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1 10NCEE Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July 21-25, 2014 Anchorage, Alaska A DIRECT METHOD FOR THE DESIGN OF VISCOUS DAMPERS TO BE INSERTED IN EXISTING BUILDINGS P.P. Diotallevi 1, L. Landi 2 and S. Lucchi 3 ABSTRACT A direct procedure to determine the supplemental damping ratio required for the retrofit of existing buildings with viscous dampers is proposed here. The method has been developed following both analytical and graphical formulations. The graphical formulation is based on the construction of constant design acceleration or constant design displacement curves. These curves allow to estimate the required effective damping as a function of the effective period, associated to the secant stiffness at maximum displacement. Combining these curves with constant ductility curves, which provide a correlation between the effective damping and the supplemental damping for given available ductility and damper typology, it is possible to obtain the required supplemental damping and to design the damping system. The effectiveness of the proposed method has then been evaluated through the comparison with nonlinear dynamic analyses of selected RC frames and the application to real case studies. In these applications, nonlinear viscous dampers have been considered for the retrofit. 1 Professor, Dept. DICAM, University of Bologna, Bologna, Italy 2 Assistant Professor, Dept. DICAM, University of Bologna, Bologna, Italy 3 Research Assistant, Dept. DICAM, University of Bologna, Bologna, Italy Diotallevi P.P., Landi L., Lucchi S. A direct method for the design of viscous dampers to be inserted in existing buildings. Proceedings of the 10 th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.

2 A direct method for the design of viscous dampers to be inserted in existing buildings P. P. Diotallevi 1, L. Landi 2 and S. Lucchi 3 ABSTRACT A direct procedure to determine the supplemental damping ratio required for the retrofit of existing buildings with viscous dampers is proposed here. The method has been developed following both analytical and graphical formulations. The graphical formulation is based on the construction of constant design acceleration or constant design displacement curves. These curves allow to estimate the required effective damping as a function of the effective period, associated to the secant stiffness at maximum displacement. Combining these curves with constant ductility curves, which provide a correlation between the effective damping and the supplemental damping for given available ductility and damper typology, it is possible to obtain the required supplemental damping and to design the damping system. The effectiveness of the proposed method has then been evaluated through the comparison with nonlinear dynamic analyses of selected RC frames and the application to real case studies. In these applications, nonlinear viscous dampers have been considered for the retrofit. Introduction In recent years the retrofit of existing buildings in order to sustain the seismic actions has become one of the most relevant problems in seismic design. The recent seismic events, as L Aquila or Emilia earthquakes in Italy, have highlighted that a relevant part of buildings is inadequate to withstand the seismic actions. As a consequence, rehabilitation interventions are required to make the structures able to satisfy the seismic requirements provided by the current code. A widespread methodology to obtain the seismic rehabilitation is the use of passive dissipation systems [1, 2, 3]. These systems allow to reduce the seismic effects on the structures by absorbing a portion of the seismic input energy and to limit the rehabilitation intervention to their addition [4, 5, 6]. Anyway the determination of the supplemental damping that must be provided still presents some difficulties [7]. Many reports [8], guidelines and international provisions [9-10] deal with the problem of the calculation of this value. In this work we refer in particular to the report published by MCEER [8]. It proposes a methodology based on the comparison between the spectral capacity curve of the structure and the design demand curve. However, in this procedure, the supplemental damping ratio is fixed a priori and the determination of the required value needs to perform iterations. 1 Professor, Dept. DICAM, University of Bologna, Bologna, Italy 2 Assistant Professor, Dept. DICAM, University of Bologna, Bologna, Italy 3 Research Assistant, Dept. DICAM, University of Bologna, Bologna, Italy Diotallevi P.P., Landi L., Lucchi S. A direct method for the design of viscous dampers to be inserted in existing buildings. Proceedings of the 10 th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.

3 This work proposes, instead, a direct procedure to determine the minimum supplemental damping ratio to be provided by the dampers for the rehabilitation of a structure for a given seismic action [11]. The procedure proposed has been developed according to both analytical and graphical approaches. The graphical formulation is based on the definition of constant design acceleration or constant design displacement curves, which can be obtained from the response spectrum, and constant ductility curves, which can be defined in relation to the typology of the dampers used for the rehabilitation. The proposed direct method has been initially studied and applied with reference to RC frames. Subsequently the methodology has been verified through comparisons with nonlinear dynamic analyses of the considered frames. Finally it has been applied for the rehabilitation of an existing building located in Italy. The proposed procedure The effective damping of a generic structure equipped with a passive dissipation system may be defined as follows: ξeff = ξi + ξh + ξv (1) where ξ i is the inherent damping ratio, ξ h is the contribution due to the hysteretic behaviour of the structural members and ξ v is the supplemental damping ratio provided by the dampers. The effective damping indicates the capability of the structure to dissipate the seismic input energy. Consequently, it is a term that indicates the possible reduction of the design acceleration due to the total dissipation. In the report MCEER the effect of dissipation is considered by associating a damping reduction factor B to each value of. This factor allows to reduce the elastic response spectrum in order to obtain the design spectrum to be used for the seismic evaluation. If this spectrum is represented in the spectral acceleration-spectral displacement plane (S a -S d plane), it is possible to obtain the demand spectrum. In order to verify if the effective damping is enough to rehabilitate the building, the demand spectrum has to be compared with the capacity spectrum of the structure. This spectrum is constructed by representing the curve obtained from the pushover analysis in the S a -S d plane and idealizing this curve with an elastic-perfectly plastic diagram. If the design demand curve intersects the spectral capacity curve, the assumed supplemental damping is enough to rehabilitate the structure. On the contrary, it is necessary to improve the damping ratio provided by the dampers. Independently from the procedure used to idealize the spectral capacity curve, it is evident that the method described requires iterations to calibrate the value of ξ v. Conversely, it is possible to define an alternative methodology that allows to obtain directly the minimum required value of the supplemental damping to rehabilitate the structure. Analytical Formulation To explain this method we refer to a generic structure with a spectral capacity curve represented by an elastic-perfectly plastic behaviour. As far as the structure is concerned, once the pushover analysis is carried out, it is possible to know the maximum acceleration bearable by the structure (S ay ), which is also the yield acceleration, the maximum spectral displacement (S dm, limit value for the considered limit state) and the available ductility. Therefore, from Eq. 1, assuming the

4 displacement demand equal to the corresponding capacity, only and ξ v remain unknown. These quantities are calculated by associating an equivalent elastic system to the structure. Its period is defined considering the line connecting the origin with the point identified by the maximum spectral displacement S dm and by the maximum acceleration S ay (Fig. 1). Sa (g) Elastic Behaviour 0.50 Spectral Demand Curve 0.45 Spectral Capacity Curve Sa,el Say 0.05 Sdm Sd (m) Figure 1. Equivalent elastic behaviour A secant period T eff is then associated to the equivalent system: T eff = 2π S S (2) dm ay where the terms are known from the spectral capacity curve. Considering now this equivalent single-degree of freedom system, it is possible to evaluate the elastic acceleration demand S a,el (Fig. 1) from the elastic response spectrum (associated to ξ= 5%). This value can be obtained from the period calculated with Eq. (2): Sa, el = Sa, el ( Teff ) (3) With the purpose to bear this acceleration it is necessary to provide a specific value of supplemental damping that allows the passage from S a,el to the maximum bearable acceleration. This value, as far as the previous considerations are concerned, is related to a reduction factor B req that is given by the ratio between the elastic spectral accelerations and the maximum one: B req S a, el = S (4) ay Once this factor is calculated, the value associated to B req can be obtained using correlation tables between B and the damping ratio [8]. Considering that: ( ξ ) B = B (5) eff can be derived as an inverse function of B req, calculated using Eqs. 3 and 4: eff ( S ( T S ) 1 ξ = B ) (6) a, el eff ay The determination of the minimum supplemental damping provided by the dampers requires to consider also the real behaviour of the structure, and in particular to evaluate the

5 dissipation due to the hysteresis of the structural members: ξh = ( 2q h π )( 1 1 μ) (7) where μ is the ductility demand, assumed equal to the ductility capacity obtained from the spectral capacity curve, and q h is a factor equal to the ratio of the actual area of the hysteresis loop to that of an elastic-plastic system. This factor is equal to 1 for elastic-plastic behaviour and it is less than 1 for loops with degradation [8]. The required supplemental damping can then be derived using Eq. 1, Eq. 7 and the obtained value of : ( ξ + ( 2 π )( 1 μ) ) ξ v = ξ eff i q h 1 (8) By substituting Eq. 6 in Eq. 8 it follows that: S 1 ael, ( T ) eff 2qh ξ 1 1 v B = ξi S + ay π μ (9) It is useful to underline that ξ v has been derived without considering the typology of dampers. Obviously the real contribution has to be evaluated in relation to the type of devices and to the damping value under elastic or inelastic response of the structure. If the structure has a nonlinear behaviour, indeed, it is necessary to explicit ξ v as a function of the supplemental damping of the same structure with linear behaviour and of the ductility demand. Considering a nonlinear structure with nonlinear fluid-viscous dampers, such relation is given by [8]: v ve ( ) 1 2 ξ = ξ μ α (10) where ξ ve is the supplemental damping for the structure with linear behaviour and α is the damper exponent. By inverting this relation and by using Eq. 9 it is possible to obtain the minimum required supplemental damping to reach the retrofit of the structure: 1 1 S 1 ael, ( T ) eff 2qh 1 ξve = ξv = B ξi 1 α α S 2 2 ay π μ ( μ) ( μ) (11) Graphical Counterpart of the Analytical Formulation The described procedure could be applied also by making reference to a graphical representation. In particular it is possible to define constant design acceleration curves (C.D.A. curves) which allow to obtain directly the effective damping for the retrofit. These curves are derived starting from the elastic response spectrum. From this spectrum and for a fixed value of the maximum bearable acceleration S ay it is possible to calculate, for every period, the ratio between the elastic spectral acceleration and the fixed one. In this way every value of T eff is associated to a value of the damping factor B and, consequently, to a value of the effective damping necessary to pass from the elastic spectral acceleration to the maximum one. Considering Eq. 6 and fixing a value of S ay, it is possible to build a curve that gives a value of as a function of T eff. Different

6 C.D.A. curves are then created by repeating this procedure for different values of the fixed acceleration S ay. With these curves it is possible to determine immediately the value of as a function of T eff and of S ay. By drawing the vertical straight line associated to T eff, the curve related to the maximum acceleration S ay is intercepted in a point that, in ordinate, gives the value of the required effective damping (Fig. 2). The C.D.A. curves can be obtained at the variation of the type of soil and of the expected seismic intensity. Note that for low values of the acceleration capacity of the building it is possible to find no values of the effective damping, since values larger than would be necessary. On the contrary, considering high values of S ay, could be lower than 5% and so not represented. Figure 2. Constant design acceleration curves It should be noted that similar curves with the same function can be constructed by making reference to the spectral displacement. Considering that: a 2 2 ( 4π ) S = T S (12) d Then, by using Eq. 12 in Eq. 6 it is possible to obtain: ξ 95% 85% 75% 65% 55% 45% 35% 25% 15% 5% T eff (s) (, ( ) ) = B S T S (13) 1 eff d el eff dm From Eq. 13 and considering different values of the displacement capacity S dm, it is possible to create the constant design displacement curves (C.D.D. curves) (Fig. 3). 95% 85% 75% 65% 55% 45% 35% 25% 15% 5% Figure 3. Constant design displacement curves Say 0.06g Say 0.1g Say 0.12g Say 0.14g Say 0.16g Say 0.18g Say 0.22g Say 0.26g Say 0.3g Sdm 0.01 m Sdm 0.02 m Sdm 0.03 m Sdm m Sdm 0.04 m Sdm 0.05 m Sdm 0.06 m Sdm 0.07 m Sdm 0.08 m Sdm 0.09 m Sdm 0.1 m T eff (s)

7 These curves correlate the displacement capacity of the structure to the effective damping required to reach the rehabilitation. The same value of can be obtained by using independently the C.D.A. curves or the C.D.D. curves. These C.D.A. or C.D.D. curves do not give any information about the supplemental damping ratio ξ v that is necessary in order to have equal to the one indicated on the graph. This result is obtained through the construction of constant ductility curves (Fig. 4). According to Eq. 1, by fixing a value of the ξ i (usually 5% of the critical one) and considering a constant ductility contribution ξ h, associated to an assigned value of μ, the effective damping changes when ξ v varies. Consequently, considering different values of the ductility, different curves associating the required supplemental damping with are constructed. Obviously, if the structure has a nonlinear behaviour, it is necessary to explicit ξ v as a function of the supplemental damping under elastic condition ξ ve and of the available ductility of the structure, as explained in Eq. 10 for nonlinear fluid-viscous dampers. The constant ductility curves (C.D. curves) for this typology of dampers with α = 0.5 are illustrated in Fig. 4. Afterward, with obtained through the C.D.A. curves or C.D.D. curves, it is possible to enter in the C.D. curves graph and read the minimum value of the required supplemental damping ξ ve at the intersection with the curve related to the available ductility of the structure. μ=1 μ=2 μ=3 μ=4 μ=5 μ=6 μ=7 μ=8 0% 0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% 17.5% 20.0% 22.5% 25.0% ξ ve Figure 4. Constant ductility curves It should be noted that the maximum value of ξ ve in the C.D. curves has been taken equal to 25% because the current Italian code [12] sets this value as the limit one. This restriction can sometimes bring no results. It may occur that the value of ξ ve that is required to rehabilitate the structure results larger than the one allowed and, consequently, it is not applicable. However it is possible to estimate the improvement of the seismic performance due to the application of the maximum allowed supplemental damping. Considering ξ ve = 25%, it is possible to calculate the maximum effective damping,max related to the available ductility of the structure using the following relation, obtained from Eqs. 1, 7 and 10: ( )( ) ( ) 1 2 ξ,max = ξ + 2q π 1 1 μ + ξ μ α (14) eff i h ve As far as this value is concerned, the structure is not able to sustain the acceleration S a,el obtained for T eff (for this period it would be necessary ), but a lower value S ael,, associated to a response spectrum relative to a lower seismic intensity. This value is calculated as follows:

8 ( ) S = S B ξ (15) ael, ay eff,max To determine this new value, it should be considered that the ratio between S a,el and S ael, is related to the difference between the required effective damping and the maximum one,max. It follows that: 1 ( eff eff,max ) B ( Sa, el Sa, el ) Δ ξ = ξ ξ = (16) As a consequence, going back to the C.D.A. or C.D.D. curves, a point can be fixed on the vertical straight line related to T eff, at the intersection with the horizontal straight line associated to Δξ. Through this point passes a curve related to the new maximum sustainable acceleration. Application of the method to simple plane frames The proposed procedure has been applied considering two RC plane frames characterized by three and six storeys (Fig. 5). These frames have been designed considering only gravity loads. 30x40 30x40 30x40 30x40 3 m 3 m 3 m 30x40 30x40 30x40 30x40 3 m 3 m 3 m 3 m 3 m 3 m 35x35 35x35 35x35 35x35 35x35 5 m 5 m 5 m 5 m 5 m 5 m 5 m 5 m Figure 5. Three and six storey frames under study The analysis of the nonlinear behaviour of the structures has been performed by using a finite element computer program [13]. In particular the plastic hinges have been characterized with a bilinear moment-rotation curve. The capacity curve of each frame has been obtained by performing a pushover analysis with a modal pattern of lateral loads. This analysis has allowed to identify the three parameters μ, S ay and S dm that are necessary to calculate the minimum supplemental damping. For the six-storey frame, referring to the collapse limit state, the following values have been obtained: μ = 3.58, S ay = g and S dm = 6.02 cm and consequently, from Eq. 2, T eff = 2.51 s. The C.D.A. curves have been constructed considering a response spectrum associated to a peak ground acceleration (PGA) equal to 0.244g. On these curves, a value of = has been obtained. Then, from the C.D. curves, a value of ξ ve = 12.6% has been determined by assuming to insert nonlinear fluid-viscous dampers with a damper exponent α = 0.5 (Fig. 6). By knowing the values of ξ ve it has been necessary to dimension the damping system. Assuming identical devices, the following relation has been used to calculate the damping coefficient C N [8]: ( 3 ( ) 2 1 N )( 2 ND = ξ 8π 2π α α λ α φ 1+ α φ 1+ α ) 1 i= 1 1 j= 1 (17) C T D m f N ve roof i i j rj

9 where T 1 is the period of the fundamental mode, f j is a displacement magnification factor which depends on the geometrical arrangement of the damper, ϕ rj1 is the relative modal displacement between the stories, N D is the number of dampers, λ is the gamma function, α is the damper exponent and D roof is the roof displacement. 95% 85% 75% 65% 55% 45% 35% 25% 15% 5% Say 0.038g Say 0.06g Say 0.1g Say 0.12g Say 0.14g Say 0.16g Say 0.18g Say 0.2g Say 0.24g Say 0.28g T eff (s) 0% 0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% 17.5% 20.0% 22.5% 25.0% ξ ve Figure 6. Calculation of using the C.D.A. curves and C.D. curves for the six-storey frame The results of the design procedure have then been verified through a series of nonlinear dynamic analyses, performed using five spectrum-compatible recorded ground motions (El Almendral, El Centro, Newhall, Petrovac and Taft). The results obtained from analyses of the frames with dampers have been compared with those of the frames without dampers. The comparison shows that the values of displacements are considerably reduced in the first configuration. Furthermore the average value of the maximum displacements obtained with each accelerogram is close to the one calculated through the pushover analysis and the design procedure (Fig. 7). In addition, a direct comparison of the configuration of the plastic hinges has shown that in the frames with dampers no plastic hinge have attained the ultimate rotation and the collapse condition (Fig. 8). This result underlines again the effectiveness of the supplemental damping calculated with the proposed method. μ=1 μ=2 μ=3 μ=3.58 μ=5 μ=6 μ=7 μ=8 Average value without dampers Pushover Average value with dampers El Almendral with dampers El Centro with dampers Newhall with dampers Petrovac with dampers Taft with dampers 3-Storey Average value without dampers Pushover Average value with dampers El Almendral with dampers El Centro with dampers Newhall with dampers Petrovac with dampers Taft with dampers 6-Storey Roof Displacement (cm) Roof Displacement (cm) Figure 7. Maximum roof displacements for the three-storey and six-storey frames Figure 8. Configuration of the plastic hinges and indication of the reached limit state (Newhall ground motion on the left, Petrovac ground motion on the right)

10 Application of the method to an existing building The proposed procedure has been applied for studying the rehabilitation of a building located in Italy. This building is characterized by a spatial RC frame structure of six storeys above ground (Fig. 9). The pushover analysis, performed for both the main directions, provided the required values (μ, S ay and S dm ) to determine ξ ve. At this point, using the C.D.A. curves constructed for a response spectrum related to the code seismic intensity, the values of have been calculated: about 85% in x direction and 95% in y direction. Considering nonlinear fluid-viscous dampers with α = 0.5, the C.D. fail to provide solutions. This means that the only insertion of the dampers is not able to rehabilitate the building. However, it has been possible to evaluate the improvement of the maximum bearable acceleration, considering ξ ve = 25%. From Eq. 14 it has been possible to calculate,max in x (58%) and in y direction (48%) and consequently Δξ in both directions. Following the procedure, it has been obtained S ael, = 0.149g in x and S ael, = 0.109g in y direction (Fig. 10a). These values are equal to 61% (x) and 47% (y) of the required values. Figure 9. Model of the existing building 95% 85% 75% 65% 55% 45% 35% 25% 15% 5% Say g Say g Say 0.1g Say 0.109g Say 0.12g Say 0.149g Say 0.18g Say 0.22g Say 0.26g % 0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% 17.5% 20.0% 22.5% 25.0% (a) T eff (s) (b) ξ ve Figure 10. Improvement of the maximum bearable acceleration (a); calculation of ξ ve using the C.D. curves for the configurations with shear walls (b) To obtain the complete rehabilitation, four additional RC shear walls have been considered on the external frames. Since the only insertion of shear walls has not been able to provide the complete rehabilitation, a damping system has been considered together with the additional walls. Using again the proposed method and considering new values of S ay, S dm and μ, a value of = has been derived for both directions. Finally, on C.D. curves, the following values of ξ ve have been obtained: 8.3% in x direction and 11.4% in y direction (Fig. 10b). μ=1 μ=1.99 μ=2.44 μ=3 μ=4 μ=5 μ=6 μ=7

11 Conclusions The proposed calculation method has proven to be a useful tool for the direct determination of the minimum required supplemental damping for the rehabilitation of existing structures. The application of the method could be analytical or graphical. The latter is based on the construction of constant design acceleration curves or constant design displacement curves and of constant ductility curves, and provides the required supplemental damping in a fast and simple way. These curves are of general validity and permit also to calculate different required damping ratios, in relation to the variation of the seismic intensity and to the type of dampers used. In addition, the application on the two RC plane frames has demonstrated a good agreement between the results of the nonlinear dynamic analysis and of the design procedure. Finally the application on the existing building has shown that for structures with low resistance capacities it is possible to define, in the design phase, the maximum degree of improvement which can be obtained considering the code limitations regarding ξ ve. References 1. Landi L, Diotallevi PP and Castellari G. On the Design of Viscous Dampers for the Rehabilitation of Plan- Asymmetric Buildings. Journal of Earthquake Engineering, 2013; 17: Benedetti A, Landi L, Merenda DG. Displacement-Based Design of an Energy Dissipating System for Seismic Upgrading of Existing Masonry Structures. Journal of Earthquake Engineering, 2014; DOI: / Landi L, Fabbri O and Diotallevi PP. A two-step direct method for estimating the seismic response of nonlinear structures equipped with nonlinear viscous dampers. Earthquake Engineering and Structural Dynamics, 2014; DOI: /eqe Soong TT, Dargush GF. Passive Energy Dissipation Systems in Structural Engineering. John Wiley & Sons, Buffalo, Constantinou MC, Soong TT, Dargush GF. Passive Energy Dissipation Systems for Structural Design and Retrofit. Monograph Series, University of New York at Buffalo, Christopoulos C, Filiatrault A. Principles of Passive Supplemental Damping and Seismic Isolation. IUSS Press, Pavia, Diotallevi PP, Landi L, Dellavalle A. A methodology for the direct assessment of the damping ratio of structures equipped with nonlinear viscous dampers, Journal of Earthquake Engineering 2012; 16 (3): Ramirez OM, Constantinou MC, Kircher CA, Whittaker AS, Johnson MW, Gomez JD, Chrysostomou CZ. Development and Evaluation of Simplified Procedures for Analysis and Design of Buildings with Passive Energy Dissipation Systems. First revision of Report MCEER , University of New York at Buffalo, BSSC. NEHRP Guidelines for the Seismic Rehabilitation of Buildings. FEMA 273, Washington D.C, BSSC. NEHRP Recommended Provisions for Seismic Regulations for New Buildings and other Structures. FEMA 450, developed by BSSC for FEMA, Washington D.C, Diotallevi PP, Landi L and Lucchi S. A direct design methodology for the seismic retrofit of existing buildings with dissipative systems. Proceedings of the Fourteen International Conference on Advances and Trends in Engineering Materials and their Applications (AES-ATEMA 2013), Toronto, Canada, 5-9August D.M. 14/01/2008. Norme Tecniche per le Costruzioni (Italian seismic code). 13. SAP2000 Non Linear, Analysis Reference Manual, Computer and Structures Inc., Berkeley, California, USA.