ESTRATEGIA DEL PLAN DEL BUEN VIVIR 2013-2017

The design procedure requires relationships between displacement ductility and equivalent viscous damping, as shown in Fig. 3.1(c). The damping is the sum of elastic and hysteretic damping:

£ e q ~ £ e l £ h y s f P*9)

where the hysteretic damping £hys/ depends on the hysteresis rule appropriate for the structure being designed. Normally, for concrete structures, the elastic damping ratio is taken as 0.05, related to cridcal damping. A lower value (typically 0.02) is often used for steel structures.

Some discussion of both components of Eq.(3.9) is required.

(a) H ysteretic Damping: Initial work on subsdtute-structure analysis (i.e. analyses

using secant, rather than initial stiffness, and equivalent viscous damping to represent hysteretic damping) by JacobsenU1!, was based on equating the energy absorbed by hysteretic steady-state cyclic response to a given displacement level to the equivalent viscous damping of the substitute structure. This resulted in the following expression for the equivalent viscous damping coefficient, %hyst\

(3J0)

m m

In Eq. (3.10),^/, is the area within one complete cycle of stabilized force-displacement response, and Fm and Am are the maximum force and displacement achieved in the stabilized loops. Note that the damping given by Eqs.(3.9) and (3.10) is expressed as the fraction of critical damping, and is related to the secant stiffness Ke to maximum response (see Fig.3.8). It is thus compatible with the assumptions of structural characterization by stiffness and damping at peak response.

Although this level of damping produced displacement predictions under seismic excitation that were found to be in good agreement^11! with time-history results for systems with comparatively low energy absorption in the hysteretic response, such as the modified Takeda rule, it was found to seriously overestimate the effective equivalent viscous damping for systems with high energy absorption, such as elasto-plastic, or bilinear rulestC2l. A reason for this can be found when considering the response of two

different systems with the same initial backbone curve (e.g. lines 1 and 2 in Fig.3.8) to an earthquake record with a single strong velocity pulse, which might be considered an extreme example of near-fault ground motion. Assume that one system has a bilinear

Chapter 3. D irect D isplacem ent-B ased D esign: F undam ental C onsiderations 77

elasto-plastic hysteresis response, while the other, with the same initial and post-yield stiffness, is bi-linear elastic. That is, it loads up and unloads down the same backbone curve, without dissipating any hysteretic energy. If the inelastic response results from a single pulse, or fling, the peak response of the two systems should be identical, since no hysteretic energy will be dissipated on the run up lines 1 and 2 to the peak response. After the peak response, the behaviour of the two systems will differ. The bi-linear elastic system will continue to respond on lines 1 and 2, while the bilinear elasto-plastic system

will unload down a different curve, and will dissipate hysteretic energy. Although real accelerograms do not consist of a pure velocity pulse, the behaviour described above is likely to form a component of the response, to a greater or lesser degree, depending on the accelerogram characteristics.

Later attempts^11’ J2J to determine the appropriate level of equivalent viscous damping were based on equating the total energy absorbed by the hysteretic and substitute - structure during response to specific accelerograms, rather than equating steady-state response to sinusoidal excitation. It is not obvious, however, that such an approach has relevance to the prediction of peak displacement response, which is the essential measure of success, or otherwise, of the substitute structure method. See \D\] for a full discussion of the development of methods relating equivalent viscous damping to ductility.

The approach adopted in this book is to use values of equivalent viscous damping that have been calibrated for different hysteresis rules to give the same peak displacements as the hysteretic response, using inelastic time history analysis. Two independent studies, based on different methodologies were used to derive the levels of equivalent viscous damping. The first involved the use of a large number of real earthquake accelerogramsi01^ where the equivalent viscous damping was calculated for each record, ductility level, effective period and hysteresis rule separately, and then averaged over the records to provide a relationship for a given rule, ductility, and period. The second study[G2J, using a wider range of hysteresis rules was based on a smaller number of

78 Priestley, C alvi and Kowalsky. D isplacem ent-B ased Seism ic D esign of Structures

analyses were separately averaged, and compared. In each case the equivalent viscous damping was varied until the elastic results of the equivalent substitute structure matched that of the real hysteretic model.

(a) Elasto-plastic (EPP)

(d) Takeda “Fat” (TF)

(e) Ramberg-Osgood (RO)

Fig.3.9 Hysteresis Rules Considered in Inelastic Time History AnalysisIG2l

The hysteresis rules considered in the second study are described in Fig.3.9. The elastic-perfectly plastic rule (Fig 3.9(a)) is characterisdc of some isolation systems, incorporating friction sliders. The bi-linear elasto-plastic rule of Fig.3.9(b) had a second slope stiffness ratio of r —0.2, and is also appropriate for structures incorporating various

C hapter 3. Direct D isplacem ent-B ased D esign: F undam ental Considerations 79

rules: Takeda Thin (Fig 3.9(c)) and Takeda fat (Fig. 3.9(d)), represent the response of ductile reinforced concrete wall or column structures, and ductile reinforced concrete frame structures respectively. Figure 3.9(e) shows a bounded Ramberg Osgood rule calibrated to represent ductile steel structures, and the flag-shaped rule of Fig.3.9(f) represents unbonded post-tensioned structures with a small amount of additional damping. Further information on these rules is provided in Section 4.9.2(g).

The two studies identified above I™*02] initially were carried out without additional elastic damping, for reasons that will become apparent in the following section. Figure 3.10 compares the resulting average relationships for an effective period of Te — 2.0 seconds for the four hysteresis rules common to both studies. It was found that the approaches resulted in remarkably similar relationships for equivalent viscous damping for all hysteresis rules except elastic-perfectly plastic (EPP), where the discrepancy was about 20%. It is felt that the difference for the EPP rule is a consequence of the use of real records, with comparatively short durations of strong ground motion in [DI], and artificial records, with longer strong ground motion durations in [G2]. It is known that the EPP rule is sensitive to record duration, as the displacements tend to “crawl” in one direction, particularly when P-A effects are included^. Both studies showed the scatter between results from different accelerograms to be greater for the EPP rule than for other rules investigated. It is likely that the results from the [G2] study will be somewhat conservative for shorter duration (i.e. lower magnitude earthquakes), but more realistic for longer duration (higher magnitude) earthquakes. In the following discussion, the average of the two studies has been used.

The Dwairi and K o w alsk ylDP study represented the hysteretic component of response in the form:

^ > s ,= c - (3.11)

where the coefficient C depended on the hysteresis rule. This has an obvious relationship to the theoretical area-based approach of Eq.(3.10) for the EPP rule, for which C ~ 2. Some period-dependency was found for effective periods Te < 1.0 seconds.

The study by Grant et aFG2', which considered a wider range of hysteretic rules, used a more complex formulation of the relationship between ductility and equivalent viscous damping, the hysteretic component of which is given by:

= a \' ~ 7

1 + 1

(Te+c)a

Equation (3.12) includes the period-dependency of the response, in the coefficients c and d. Table 3.1 lists the coefficients for the various hysteresis rules investigated.

Eq

uiv

. V

isc

ou

s

Da

mp

ing

Eq

uiv

.V

isc

ou

s

Da

mp

ing

Ra

tio

80 P riestley, Calvi and Kowalsky. D isplacem ent-B ased Seism ic D esign of Structures

Displacement Ductility