Ductility-dependent equivalent damping for displacement-based design (DDBD)

2.2.2.1 Introduction

The concept of viscous damping is generally used to represent the energy dissipated by the structure in the elastic range. Such dissipation is due to various mechanisms such as cracking, nonlinearity in the elastic phase of response, interaction with non-structural elements, soil-structure interaction, etc. As it is very difficult and unpractical to estimate each mechanism individually, the elastic viscous damping represents the combined effect of all of the dissipation mechanisms. There is no direct relationship of such damping with the real physical phenomena. However, the adoption of the viscous damping concept facilitates the solution of the differential equations of motion, represented by Eq. (2.24):

0

2 + ² =

+ u u

u ξϖ_n ϖ_n ^(2.24a)

where

m n

c ξ ϖ

= 2 (2.24b)

u is the displacement, ωn is the natural vibration frequency (radians/sec) of the system and ξ is the damping ratio or fraction of critical damping.

As mentioned above, the assumption of viscous damping simplifies greatly the dynamic problem and this is the reason why in the direct displacement based design the non-linear behaviour has been also represented by including the hysteretic damping into the damping term of Eq.(2.24a). Using this approach it is possible to solve a simple linear system instead of a non-linear system which is more time and resource demanding for design applications.

0 5 10 15 20 25 30 35 40 45 50

1 2 3 4 5 6 7

Ductility

Equivalent Damping Factor

Rosenblueth Steel*

Conc. beam * Conc. Wall*

Kow alsky Gulkan Iw an

Prestr. Conc.*

Figure 2.26 Equivalent viscous damping factor for Eq. (2.24) * proposed by Priestley [2003].

2.2.2.2 Existing equivalent viscous damping equations

There are multiple references which report different equations for the equivalent viscous damping factor, including those by Priestley[2003], Fardis and Panagiotakos[1996], Miranda and Ruiz [2002], Calvi [1999]. Some are based on Jacobsen’s approach [1930, 1960], others on the substitute damping approach, and others on results of time-history analyses. Some of the reported equations are compared in Figure 2.26.

Miranda [2002] carried out an investigation comparing the capabilities of different performance based methodologies (including methodologies based on equivalent linearization) in estimating the inelastic displacement for Takeda, modified Clough, stiffness degrading and elastoplastic system using 264 ground motion records. Similarly to many other researchers [e.g. Gulkan and Sozen, 1974; Iwan, 1980; Judi et al., 2000;

Kowalsky and Ayers, 2002; Priestley, 2003], the findings of Miranda also suggested that Jacobsen’s approach for estimating the equivalent viscous damping factor was non conservative for structures with high hysteretic energy absorption.

This study therefore aims at carrying out additional research in order to complement previous studies and to optimise damping values applicable to a wide range of hysteretic models and period ranges. For the sake of brevity, however, only a summary of the results obtained will be included here, the reader being instead referred to other publications [e.g. Grant, 2005; Priestley and Grant, 2005; Priestley et al., 2007]

2.2.2.3 Methodology for calibrating equivalent viscous damping for DDBD A methodology was implemented, to modify, where necessary, Jacobsen’s equations for

the equivalent viscous damping factor, ξ. The procedure adopted determined the value of equivalent damping that has to be applied to an equivalent elastic system with a given effective period (based on the secant stiffness to maximum displacement response) in order to match its response (in terms of maximum displacement) to that obtained from a system with the same period (effective period) and a given level of ductility using nonlinear time-history analysis. The final objective of this procedure was to develop equations that define the equivalent damping factor to be used in DDBD for a given level of ductility and hysteretic model. Six hysteretic models, depicted in Figure 2.27 were considered in the analysis:

• Modified Takeda Model [Loeding et al., 1998], employed in two different versions in the analyses. The first, a “thin” Takeda model with α _{=0.5 and} β =0 was considered appropriate for bridge piers and wall structures. The second “fat” Takeda model (α

=0.3, β =0.6) was intended to be appropriate for reinforced concrete frames. In both cases the post-yield stiffness ratio was taken as 0.05.

• Bilinear [Otani, 1981], intended to model a bridge structure isolated with friction pendulum dampers. The behaviour represented the composite flexibility of dampers plus bridge piers, and a high post-yield stiffness ratio of r = 0.2 was consequently adopted.

• Elastic-Perfectly Plastic [Otani, 1981], included solely because of its historic importance in seismic time-history analysis. Its closest approximation in real structures is a flexible structure isolated with a flat coulomb (friction) damper.

• Ramberg Osgood [Otani, 1981], selected as being reasonably appropriate for structural steel members.

• Ring-spring, chosen to represent a precast concrete structure connected with unbonded prestressing, resulting in a hysteretic model characterized by bilinear elastic response with low hysteretic damping.

Six different synthetic accelerograms were selected in order to carry out the time-history analysis of the SDOF systems. All of them were constructed matching a given displacement design response spectrum at 5% damping. The first accelerogram was a synthetic record based on a record from Manjil, Iran, 20^th June 1990, developed by Bommer and Mendis [2005] for use specifically in this research. The rest of the accelerograms were artificial records obtained as part of this study and complemented with others obtained by Alvarez [2004] and Sullivan [2003]. Average spectra for 20 levels of damping were computed for values between 3% and 60%. The spectral value for a given damping was obtained by interpolation between the closest upper and lower spectra bounding it. All of the records were compatible with the ATC32 design spectrum for soil type C and PGA 0.7 g. [ATC, 1996]. The records were selected so that the results of the dynamic analyses are representative of a larger set of non-spectrum compatible records.

dp

Equivalent viscous damping in DDBD is defined as the combination of two effects: the elastic damping and the hysteretic damping. The initial elastic viscous damping used for time–history analysis of SDOF systems has been traditionally defined in practice by use of a constant damping coefficient corresponding to 5% of critical damping, though lower values are sometimes used for steel structures. This value is assumed to represent the different sources of energy dissipation when the structure is considered in the elastic range. It is not clear that constant coefficient damping is appropriate for structures

responding inelastically, since the hysteretic models generally incorporate the full structural energy dissipation in the inelastic range, and other contributory mechanism, such as foundation damping will be greatly reduced when the structure enters the inelastic range. It would appear that tangent-stiffness proportional damping would be more appropriate than constant coefficient (initial-stiffness proportional, or mass-proportional) damping in modelling initial elastic damping in seismic response. The adoption of different characteristic stiffness in DDBD (secant stiffness) and time-history analyses (initial stiffness) further confuses the issue.

The representation of elastic damping is often given little consideration, but as shown in Priestley and Grant [2005], the issue of how this damping should be represented is important, not just to DDBD, but also to time-history analysis. In this study the influence of elastic damping was removed from both the design process and the time-history validation, by specifying zero elastic damping.

Time-history analysis were carried out using the program Ruaumoko [Carr, 1996] using a Newmark constant average acceleration integration scheme with β = 0.25. As described above, this procedure was carried out iteratively until the displacement of the equivalent SDOF system was the same for the time-history and for the design spectral analysis. The time step used for the integration was taken as half of the discretization step of each accelerograms, this is, 0.005 seconds except for the Manjil adjusted record which has a discretization step of 0.0045.

Table 2.6 Constant values for equivalent viscous damping equation Constant Takeda

Thin Takeda

Fat Bilinear EPP Ramberg

Osgood Ring Spring a 95 130 160 140 150 50 b 0.5 0.5 0.5 0.5 0.45 0.5

c 0.85 0.85 0.85 0.85 1 1

d 4 4 4 2 4 3

2.2.2.4 Design equations for equivalent viscous damping factor

The equations proposed by Priestley [2003] were taken as the base for the modified equations proposed here. The resulting equations have the following form:

(

)

a r

eff d effective b

1 1 1

1 . 1 0

1 ⎟⎟⋅

⎠

⎞

⎜⎜

⎝

⎛ + +

⎟⎟⋅

⎠

⎜⎜ ⎞

⎝

⎛ − − ⋅ ⋅

⋅

= µ

µ

ξ π ^(2.25)

where a, b, c and d are coefficients defined for each hysteretic model, µ is the ductility, Teff

is the effective period, r is the post–elastic stiffness coefficient (applicable only in the case of the bilinear model) and N is a normalising factor. An important difference of this equation from previous existing proposals is the extra term which is dependent on effective period. An exception exists in the recommendations of Judi et al. [2002], which indicate weak period-dependency.

The process to obtain the calibration factors a, b, c, d was carried out for each hysteretic model analyzed. Not all the correction factors depend in the same proportion on the variables (period and ductility). A perfect match was not possible for all the cases because it was necessary to keep a simple form of the equation. Table 2.6 shows the constants obtained for the hysteretic models analyzed.

It is important to mention that the implementation of this approach in DDBD will modify slightly the design process, since in the latter the damping is obtained directly from the ductility; then, the effective period is obtained for a given target displacement.

However, using the modified equation it will be necessary to iterate in order to obtain the period. The additional work is, however, insignificant.

One alternative to this procedure could be derived from the fact that the dependency of the equivalent damping reduces very rapidly as the period increases. The damping becomes almost independent for a period longer than 1 sec. Using this approximation it would not be necessary to iterate in the design procedure as mentioned previously.

However, given that the procedure is simple the proposed equation has the advantage of being general for any period. Reduced versions of the equation may be obtained easily according to the characteristics of the structures considered in the design.

2.2.2.5 Concluding remarks

The design displacements based on Jacobsen’s approach within the direct displacement-based design methodology were calculated for six different hysteretic models with different effective periods and displacement ductilities. These design displacements were compared with results from time-history analyses using six earthquake records compatible with the design spectra. Both designs and time-history analyses were carried out using zero elastic viscous damping to enable the contribution of hysteretic damping to be directly determined. As has been found in earlier studies, the results obtained were inconsistent, with the time-history displacements exceeding the design displacements in many cases, particularly for hysteretic models with high energy absorption.

An iterative procedure was used to determine the required value for equivalent viscous

damping to be used in direct displacement-based design to equate the design displacement and the time-history results. From these analyses a series of design equations determining the equivalent viscous damping as a function of hysteresis rule, displacement ductility and period were developed.

3 ESTIMATION OF COMPONENT FORCE AND DEFORMATION CAPACITIES IN BUILDINGS