VISCOUS DAMPING IN SEISMIC DESIGN AND ANALYSIS

Transcription

1 Journal of Earthquake Engineering Imperial College Press VISCOUS DAMPING IN SEISMIC DESIGN AND ANALYSIS M. J. N. PRIESTLEY and D. N. GRANT European School for Advanced Studies in Reduction of Seismic Risk (ROSE School), Via Ferrata, Pavia, Italy Received September 2004 Revised Accepted The characterisation of viscous damping in time history analysis is discussed. Although it has been more common in the past to use a constant damping coefficient for single-degree-of-freedom time history analyses, it is contended that tangent-stiffness proportional damping is a more realistic assumption for inelastic systems. Analyses are reported showing the difference in peak displacement response of single-degree-of-freedom systems with various hysteretic characteristics analysed with 5% initial-stiffness or tangent-stiffness proportional damping. The difference is found to be significant, and dependent on hysteresis rule, ductility level and period. The relationship between the level of elastic viscous damping assumed in time-history analysis, and the value adopted in Direct-Displacement Based Design is investigated. It is shown that the difference in characteristic stiffness between time-history analysis (i.e. the initial stiffness) and displacement based design (the secant stiffness to maximum response) requires a modification to the elastic viscous damping added to the hysteretic damping in Direct Displacement Based Design. Keywords: Seismic analysis; viscous damping; displacement based design; equivalent linearisation. 1. Introduction Over the past 45 years there have been many studies [e.g. Veletsos and Newmark, 1960; Iwan, 1980; Miranda, 2000] investigating the response of structures with different hysteretic rules to earthquake records using inelastic time-history analyses. Initially, the purpose of these studies was to determine the required force reduction factors, R, (or behaviour factors q) to be used in force-based seismic design that would result in a specified displacement ductility µ being achieved. Typically, it was found that the results were rather insensitive to the hysteretic rule adopted, and that for initial (elastic) periods greater than about 0.5 seconds the equal-displacement approximation, which states that R = µ, provides an adequate approximation of behaviour. For periods shorter than about 0.5 seconds, R < µ was required, with the value of R decreasing as the period reduced. These findings agreed with the earlier recommendations of Newmark and Hall [1982]. More recently, a number of studies [e.g. Judi et al., 2000; Miranda and Ruiz-Garcia, 2002] have investigated the response of inelastic systems with different hysteretic rules with the specific intent of determining the relationship between ductility and equivalent viscous damping for use in Direct Displacement-Based Design (DDBD) [Priestley, 2000]. As shown in Fig. 1, DDBD approximates a multi-degree-of-freedom (MDOF) 1

2 2 M. J. N. Priestley & D. N. Grant structure by an equivalent single-degree-of-freedom substitute structure (SDOF), (Fig. 1(a)), and characterises the seismic response by the secant stiffness to maximum displacement response (Fig. 1(b)), and equivalent viscous damping representing the combined effects of elastic and hysteretic energy dissipation (Fig. 1(c)). With knowledge of the design displacement and the equivalent viscous damping, the effective period at peak displacement response can be found from a set of displacement spectra for different levels of equivalent viscous damping, and hence the required secant stiffness to maximum displacement response, and required strength, can be directly obtained [Priestley, 2000]. One of the key research areas in DDBD over recent years has been the analysis of SDOF structures to refine the relationships between hysteretic model and equivalent viscous damping. A companion paper [Blandon et al., 2005] reports recent results for a number of hysteretic models representing a wide range of structural types. An aspect of DDBD that has not received much attention is the characterisation of the initial elastic viscous damping, where elastic is used in this paper to distinguish from the equivalent viscous damping used in the substitute structure approach. In the past, as is common in many areas of seismic engineering, it has been assumed that a value of 5% initial elastic damping is appropriate, and should be added to the hysteretic damping, as indicated in Fig. 1(c). Thus, the fraction of critical damping, ξ eq, for the linear substitute structure is given as a function of the target displacement ductility, µ, by an expression of the following form: ξ eq ξeq, v + ξeq, h, µ =, (1) where ξ eq,v and ξ eq,h,µ represent energy dissipation resulting from viscous damping and inelastic hysteresis, respectively, with ξ eq,v generally assumed to be 5%, as noted above. However, it is apparent that there is a possible inconsistency here. In DDBD the 5% initial elastic damping is related to the secant stiffness to maximum displacement, whereas it is normal in SDOF dynamic analysis to relate the elastic damping to either the initial (elastic) stiffness, or less commonly, a stiffness that varies as the structural stiffness degrades with inelastic action (tangent stiffness). Although the terminology is inexact, since the damping coefficient is proportional to the square root of the stiffness, we will term these initial stiffness proportional and tangent stiffness proportional damping, as they are analagous to damping assumptions used for MDOF dynamic analysis. Since the response velocities of the real and substitute structures are expected to be similar under seismic response, the damping force, which is equal to the product of the damping coefficient and the velocity, will differ significantly, since the effective stiffness k eff of the substitute structure is approximately equal to k eff = k i / µ (for low post-yield stiffness). It would appear that some adjustment would be needed to the value of initial elastic damping assumed in DDBD to ensure compatibility between the real and substitute structures. Without such an adjustment, the verification of DDBD by inelastic time-history analysis would be based on incompatible assumptions. The discussion in the previous paragraph begs the question of how to characterise initial elastic damping in inelastic time-history analysis (ITHA). Typically research papers reporting results on SDOF ITHA state that 5% initial elastic damping was used,

3 Equivalent Viscous Damping in Seismic Design and Analysis 3 without clarifying whether this has been related to the initial or tangent stiffness. This is because in SDOF analyses, no special damping model is required, and the damping coefficient is assumed to be constant. With MDOF analyses, the situation is often further confused by the adoption of Rayleigh damping, which is a combination of mass proportional and stiffness proportional damping. It is believed that many analysts consider the choice of initial elastic damping model to be rather insignificant for either SDOF or MDOF inelastic analyses, as the effects are expected to be masked by the much greater energy dissipation associated with hysteretic response. This is despite evidence by others [e.g. Otani, 1981] that the choice of initial damping model between a constant damping matrix (derived through the assumption of mass proportional or initial-stiffness proportional damping) and tangent-stiffness proportional damping matrix could be significant, particularly for short period structures. Note that a more rigorous mathematical discussion of damping models for SDOF systems is presented in Section 4.1 below. This paper investigates the significance of the choice of SDOF initial elastic damping model to displacement response of different hysteretic models in some detail, and establishes the relationship between initial elastic damping models in real and substitute structures. 2. Choice of Initial Elastic Damping Model As noted above, it appears that the vast majority of analysts use initial-stiffness proportional elastic damping for SDOF analysis, without considering whether or not this is appropriate. It is believed that this is based on elastic response measurements which indicate that a constant level (typically 5% for concrete structures, but often less for steel structures) is appropriate for the elastic range of response. It is not clear, however, that this is appropriate for inelastic response. There appear to be three main reasons for incorporating initial elastic damping in ITHA: The assumption of linear elastic response at force-levels less than yield: Many hysteretic rules make this assumption, and therefore do not represent the nonlinearity, and hence hysteretic damping within the elastic range for concrete and masonry structures, unless additional damping is provided. Foundation damping: Soil flexibility, nonlinearity and radiation damping are not normally incorporated in structural time-history analyses, and may provide additional damping to the structural response. Nonstructural damping: Hysteretic response of nonstructural elements, and relative movement between structural and nonstructural elements in a building may result in an effective additional damping force. Discussing these reasons in turn, it should be recognised that hysteretic rules are generally calibrated to structural response in the inelastic phase of response. Therefore additional elastic damping should not be used in the post-yield state to represent

4 4 M. J. N. Priestley & D. N. Grant structural response except when the structure is unloading and reloading elastically. If the hysteretic rule models the elastic range nonlinearly then no additional damping should be used in ITHA for structural representation. It is thus clear that the elastic damping of hysteretic models which have linear elastic branches, and hence do not dissipate energy by hysteretic action at low force levels would be best modelled with tangent-stiffness proportional damping, since the elastic damping force will greatly reduce when the stiffness drops to the post-yield level. It should, however, be noted that when the post-yield stiffness is significant, the elastic damping will still be overestimated. This is particularly important for hysteretic rules such as the modified Takeda degrading stiffness rule which has comparatively high stiffness in post-yield cycles. If the structure deforms with perfect plasticity, then foundation forces will remain constant, and foundation damping will cease. It is thus clear that the effects of foundation damping in SDOF analysis are best represented by tangent stiffness related to the structural response, unless the foundation response is separately modelled by springs and dashpots. It is conceivable that the non-structural damping force is displacement-dependent rather than force-dependent, and hence a constant damping coefficient may be appropriate for the portion of elastic damping that is attributable to non-structural forces. There are two possible contributions to non-structural damping that should be considered separately: Energy dissipation due to hysteretic response of the nonstructural elements Energy dissipation due to sliding between nonstructural and structural elements For a modern frame building, separation between structural and non-structural elements is required, and hence they should not contribute significantly to damping. Further, even if not separated, the lateral strength of all non-structural elements is likely to be less than 5% of the structural lateral strength (unless the non-structural elements are masonry infill). If we assume 10% viscous damping in these elements, an upper bound of about 0.5% equivalent viscous damping related to the structural response seems reasonable. Nonstructural elements are unlikely to play a significant role in the response of bridges. Sliding will normally relate to a frictional coefficient, and the weight of the nonstructural element. Unless the non-structural elements are masonry, the frictional force is likely to be negligible. It should be noted that it is probable that so-called nonstructural infill initially contributes more significantly to strength, stiffness, and damping than is the case with (e.g.) lightweight partitions. However, it is known that the strength degrades rapidly for drift levels > 0.5% (which is generally less than structural yield drift). The damping force is also likely to degrade rapidly. The effects of non-structural masonry infill should be modelled by separate structural elements with severely degrading strength and stiffness not by increased viscous damping.

5 Equivalent Viscous Damping in Seismic Design and Analysis 5 It is thus recommended that for modern buildings with separated or lightweight nonstructural elements, viscous energy dissipation should be modelled by a tangent-stiffness proportional damping coefficient. Note that the specification of tangent-stiffness proportional Rayleigh damping for MDOF, multi-storey buildings will not have the desired effect, since most of the elastic damping in the critical first mode will be mass proportional, which is constant with inelastic action. Consider the basic form for the fraction of critcial damping in the Rayleigh damping model: 1 α ξ = + βωn 2 ω n, (2) where α and β are the coefficients associated with mass proportional and stiffness proportional damping respectively, and ω n is the circular frequency. If we specify the same value (say 5%) for ξ at two different frequencies, where the higher one is κ times the lower (fundamental) frequency, then: and 1 α ξ = + βω1, (3) 2 ω 1 1 α ξ = + βκω1, (4) 2 κω 1 Subtracting Eq. (4) from κ times Eq. (3): ξ α 2ω κ 1 κ ( κ 1) = 1 2, (5) and hence the damping α / 2ω1 associated with mass proportional damping in the first mode is ξ κ /( κ +1), while the damping attributed to stiffness proportional damping in the first mode is ξ /( κ +1). Consider the case where we specify 5% damping at T 1 = 1.5 sec and T 2 = 0.3 sec. Hence κ = 5. Then, even if we specify tangent-stiffness proportional Rayleigh damping, only 0.83% is stiffness proportional, while 4.17% is mass proportional, and hence acts in an identical manner to initial-stiffness damping when the structure responds inelastically. 3. Significance of Choice of Damping Model 3.1. Steady-state response It is instructive in determining the influence of alternative elastic damping models to consider the steady-state response of an inelastic SDOF oscillator subjected to sinusoidal excitation. This enables direct comparison between hysteretic and elastic damping energy dissipation, and also between elastic damping energy using a constant damping coefficient and tangent-stiffness proportional damping models. To this end a simple

6 6 M. J. N. Priestley & D. N. Grant SDOF oscillator with initial period of 0.5 sec and a modified Takeda hysteresis rule for inelastic response typical of a concrete bridge pier was subjected to 10 seconds of a 1.0 Hz forcing function. Details of the Takeda model which included a post-yield stiffness equal to 5% of the initial stiffness are discussed later in relation to Fig. 3(a). Flexural action of the bridge pier was modelled by axial deformation of a linear spring, using the ITHA program Ruaumoko [Carr, 2001]. The constant damping coefficient case was specified by selecting initial-stiffness proportional damping in the program, although for SDOF analysis, the results will be identical to the mass proportional case. As noted above, initial-stiffness proportional is a misnomer for SDOF analysis, as the damping coefficient, and thust the damping force, is actually proportional to the square root of the initial stiffness (see Eq. (8)). The steady-state response of the pier corresponded to a displacement ductility of about 7.7 at the upper limit of reasonable ductile response. Results for the stabilised loops, ignoring the transitory first three seconds of response are plotted in Fig. 2(a) (initial-stiffness proportional damping) and Fig. 2(b) (tangentstiffness proportional damping). In each case the hysteretic response associated with nonlinear structural response is plotted on the left, and the elastic damping forcedisplacement response is plotted to the right. The areas inside the loops indicate the relative energy absorption. For the case with initial-stiffness proportional damping, the energy absorbed by elastic damping is approximately 83% of the structural hysteretic energy dissipation, despite the high ductility level. This might be surprising when it is considered that the elastic damping corresponds to 5% of critical damping, while the hysteretic damping is equivalent to about 20% of critical damping. This anomaly is due to the different reference stiffness used. The elastic damping is specified with respect to the initial stiffness, whereas the hysteretic damping is calculated from the secant stiffness to maximum response. When the elastic damping is tangent-stiffness proportional, as we believe to be most appropriate for structural response, the elastic damping area is greatly reduced, as can be seen by comparing the upper and lower right-hand plots of Fig. 2. In the lower plot, the reduction in damping force corresponding to the stiffness change is clearly visible. In this case, the area of the elastic damping loop is only about 15% of the structural hysteretic energy dissipation Response to Earthquake Records Hysteresis models It is clear from the simple comparison of steady-state response that there is a significant difference between the total energy absorbed (hysteretic plus elastic damping) using a constant damping coefficient and a coefficient that is proportional to the tangent-stiffness, for steady-state response. It is not clear, however, that this will translate into significant differences in peak displacement response under transitory seismic response. In order to investigate this aspect, a series of analyses were carried out for SDOF oscillators with initial periods between 0.25 seconds and 2.0 seconds, using the six different hysteretic

7 Equivalent Viscous Damping in Seismic Design and Analysis 7 models of Fig. 3. Two modified Takeda [Otani and Sozen, 1972] models representative of reinforced concrete bridge piers were considered with post-yield stiffnesses of 0.2% and 5% of the initial stiffness, respectively. The post-yield stiffness ratios represent reasonable lower and upper bounds. Unloading stiffness was given by: γ k = / µ, (6) u k i where γ is the Takeda unloading coefficient, taken equal to 0.5, and µ is the displacement ductility factor. On reloading, the rule points to the previous peak force-displacement response. Two bilinear hysteretic models were investigated, again with post-yield stiffnesses of 0.2% (essentially elastic-perfectly plastic) and 5%, respectively. Unloading and reloading stiffnesses were equal to the initial stiffness. The bilinear rule can be considered an upper bound of hysteretic energy dissipation for common structural materials. The final two hysteretic models followed a flag hysteresis rule (Fig. 3(c)), with 5% post-yield stiffness in both cases. The flag hysteresis rule is representative of unbonded post-tensioned structures with additional damping [Stanton et al., 1998], with the area inside each lobe of the loop dependent on the amount of additional energy dissipation provided. For these analyses, values of the loop area parameter (see Fig. 3(c)) of β = 0.35 and 0.7 were chosen. Note that this loop has zero residual displacement. A measure of the hysteretic damping of the different models can be obtained from the equivalent viscous damping of the stabilised steady-state response, related to the secant stiffness to maximum response, an approach first carried out by Jacobsen for nonlinear viscous [Jacobsen, 1930] and hysteretic [Jacobsen, 1960] models. For the three hysteretic types considered, the following equivalences apply: 1 π Takeda model: ξ = 1 ( r) Bilinear model: Flag model: γ 1 γ ( µ r ) eq, h, µ 1 µ ( µ 1)( 1 r) ( 1+ r( µ 1) ) ; (7a) 2 eq, h, = ; (7b) πµ ξ µ ξ, h, µ ( π 1) β ( 1+ r( µ 1) ) eq =. (7c) πµ These expressions are plotted in Fig. 4(a) for the different values of r (post-yield stiffness ratio), displacement ductility µ, and β (flag loop parameter). Figure 4(b) plots the ratio of effective period at maximum displacement response to initial period as a function of r and displacement ductility Earthquake records A basic design spectrum corresponding to the ATC32 [Applied Technology Corporation, 1996] Soil Category C, with a peak ground acceleration of 0.56g was selected for the study, though this choice was arbitrary. Five spectrum-compatible records were generated with Simquke [Carr, 2001], each with a duration of 30 seconds. In addition, a real earthquake record (El Centro 1940 NS) was scaled to give approximately the same

8 8 M. J. N. Priestley & D. N. Grant displacement as the design spectrum at the initial period as a reality check on the results from the artificial records. Elastic displacement spectra for 5% damping for the five artificial records, their average, and the design displacement spectrum are compared in Fig. 5. There is a tendency to use real earthquake records for parameter studies of this kind, as being more representative of expected response than will be the case with artificial records. In such cases the real accelerograms are scaled to match the design spectrum at the initial period of the structure analysed. However, unless the accelerogram also matches the design spectrum at the degraded effective period at maximum displacement response, the results will not be appropriate for the design spectrum. Use of a large number of real accelerograms will not improve this, unless the average spectrum matches the design spectrum over the full range of periods. In the longer period range (say from 1.5 to 4 seconds), the real records need to be chosen carefully as individual records tend to have acceleration spectra that decrease more rapidly than given by typical uniform risk spectra. For this reason, and to reduce the number of analyses necessary, the bulk of the study in this section was carried out using artificial records, although the results from the El Centro record, which showed greater scatter, but similar trends, were included in the averaged results Analysis methodology For each initial vibration period considered in the analyses, elastic analyses with 5% damping were initially run, and the average of the peak displacements, and corresponding maximum forces from the six records, were determined. A series of inelastic analyses were then run for the six different hysteretic rules using force-reduction factors of R = 2, 4 and 6, with both a constant damping coefficient (initial-stiffness proportional damping) and tangent-stiffness proportional elastic damping. The resulting averaged peak displacements were then compared, and also compared with the elastic displacement. All analyses were carried out using Ruaumoko [Carr, 2001], using a time-step of seconds Results of analyses Figure 6 shows a typical comparison of the displacement response for a SDOF oscillator with initial-stiffness and tangent-stiffness proportional elastic damping. In this example the El Centro record has been used, the initial period was 0.5 seconds, a Takeda hysteretic rule with r = 0.05 was adopted, and the force-reduction factor was approximately 4. The peak displacement for the tangent-stiffness proportional elastic damping case is 44% larger than for the initial-stiffness proportional damping case, indicating a very significant influence. Complete results from the parameter study are presented in Table 1, and in Figs In Table 1, the average results for a given elastic period, force-reduction factor R, and elastic damping model have been divided by the average elastic displacement for the same period. Tangent-stiffness and initial-stiffness proportional results are indicated by

9 Equivalent Viscous Damping in Seismic Design and Analysis 9 T and I respectively, and the number following T or I represents the force-reduction factor used in the analysis. Table 1 also includes the ratio of tangent-stiffness displacement to initial-stiffness displacement (e.g. T4/I4 indicates the displacement ratio for a force-reduction factor of 4). Figures 7, 8 and 9 plot the relationship between the displacement ratio / elastic for different force-reduction factors, for Takeda, Bilinear and Flag hysteresis rules respectively. Results for the tangent-stiffness and initial-stiffness elastic damping assumptions have been plotted with solid and dashed lines respectively. The equaldisplacement assumption that / elastic = 1.0 has also been plotted as a bold straight line, for comparison. Note that in these figures, values of / elastic for T = 0.25 sec, which it will be seen from Table 1 are frequently very high, have been omitted when greater than about 2.7, so that the trends at larger periods are more apparent. From these plots, and Table 1, the following conclusions can be drawn: Use of tangent-stiffness proportional damping always results in larger peak displacements than use of initial-stiffness proportional damping, for all hysteresis rules considered. The difference between tangent-stiffness and initial-stiffness displacements increases as the force-reduction factor R increases. This implies an increase in the difference as the displacement-ductility increases, as would be expected. The difference between tangent-stiffness and initial-stiffness displacements tends to reduce as the post-yield stiffness ratio r increases. This effect is, however, less significant than the reduction in displacement of both tangent-stiffness and initialstiffness results with increase of r. The Bilinear hysteresis models show the smallest difference between tangentstiffness and initial-stiffness displacements, but the difference is still significant, particularly at moderate to high force-reduction (or ductility) factors. The difference is particularly pronounced for the Flag hysteresis model with β = 0.35 (low additional damping), and high ductility. This is of particular significance to prestressed structural systems with unbonded tendons and low, or zero additional damping, as it is common for these systems to be designed for high ductility demand as a consequence of their low damage at moderate to high ductility levels. The equal-displacement approximation is reasonable only if the assumption of initial-stiffness damping is made, and then only for force-reduction factors of R < 4, and periods greater than 0.5 seconds. The equal-displacement approximation is non-conservative when tangent-stiffness elastic damping is assumed, except with Takeda hysteresis, R = 2, and Bilinear hysteresis, r = In virtually all other cases the equal-displacement approximation is significantly non-conservative. As noted above, Figs. 7 to 9 plot the results of the tangent-stiffness and initial-stiffness analyses as separate lines. In order to facilitate a quantitative examination of the effect of the choice of damping model, the T/I results of Table 1 have been plotted for the three types of hysteresis model in Fig. 10. For Takeda hysteresis, the tangent-stiffness

10 10 M. J. N. Priestley & D. N. Grant displacements are about 10%, 20%, and 35% higher than initial-stiffness displacements, on average, for force-reduction factors of 2, 4, and 6 respectively; for Bilinear hysteresis, the ratios are approximately 4%, 8%, and 15%, while for Flag hysteresis the average ratios are about 20%, 30% and 50%. Generally the ratios decrease as the period increases. It is felt that, except for Bilinear with low force-reduction factors, these ratios are too large to be ignored in ITHA. 4. Elastic Viscous Damping in DDBD As discussed in Sec. 1, existing applications of the substitute structure approach in displacement-based design methods assume a constant value of ξ eq,v (see Eq. (1)), equal to the fraction of critical damping, ξ, assumed appropriate for the initial elastic response. The same value is normally assigned in the ITHA used to verify the design. In the equation of motion, however, it is the damping coefficient, c, that is multiplied by the velocity to obtain the viscous damping force, and therefore the viscous energy dissipation. If the energy dissipation is to be the same in the inelastic model and the substitute structure, then the fraction of critical damping assigned in design should depend on the assumed relationship between c and ξ (see Secs , below), and will, in general, not be identical to ξ. Furthermore, for damping models in which c changes with inelasticity, the appropriate value of ξ eq,v will depend on the ductility. The relationship between damping force, F d, and displacement, for steady-state harmonic response under Bilinear hysteresis is illustrated in Fig. 11 for three damping models: a constant damping coefficient, tangent-stiffness proportional damping, and Rayleigh damping with a 50% mass and 50% tangent stiffness contribution at the system period of vibration. As discussed below, Rayleigh damping is not commonly used in SDOF analyses, as it requires at least two modes of vibration to fully determine the coefficients of the model. Points labelled Y and U represent changes in stiffness due to yielding and unloading, respectively, and the subsequent changes in the damping coefficient for the latter two models. For each model, the specified fraction of critical damping, ξ, is identical, but clearly the energy dissipation, represented by the area contained in the viscous hysteresis loops, is model-dependent. In addition, the substitute structure model for which ξ eq,v = ξ has been specified is shown in the same figure. In this case, the damping coefficient and the stiffness of the equivalent linear hysteresis model are constant, so the shape of the damping force-displacement loop is elliptical for steady-state harmonic response. It is clear from Fig. 11 that ξ eq,v = ξ will not result in the same viscous energy dissipation as the ITHA, regardless of hysteresis model and damping model. For DDBD applications, it will be important to obtain appropriate values of ξ eq,v for hysteresis models and damping models representative of real structural response. This is of particular importance for verifying DDBD with ITHA, as in this case the models are known exactly, and this source of error can be effectively removed from the comparison. For actual design applications, there is more uncertainty in the modelling, particularly for

11 Equivalent Viscous Damping in Seismic Design and Analysis 11 the viscous damping. In this case, we believe that it is appropriate to use values corresponding to tangent stiffness proportional damping, as discussed in Sec Analytical method for viscous damping in DDBD Figure 11 suggests that a procedure similar to Jacobsen s approach can be used to obtain correction factors, λ, for DDBD, such that ξ eq,v = λ ξ, to provide equal viscous energy damping in the real and substitute structure under steady-state response. We compute the viscous energy dissipated in a single cycle of steady-state harmonic response in a nonlinear SDOF system, with ξ specified for a given damping model, and equate it with the energy dissipated by a substitute structure with stiffness k eff, and damping ξ eq,v. The results from this procedure for Bilinear and modified Takeda hysteresis are summarised in the following subsections. The Takeda unloading parameter, γ, is equal to 0.5, and the β-parameter, introduced in some applications of the modified Takeda model [Carr, 2001] is equal to zero; generalised results for γ 0. 5 and β 0, and more details of the derivations can be found in Ref. [Grant et al., 2004] Constant damping coefficient As discussed in Sec. 1, a damping coefficient that does not change under inelastic response is usually adopted in SDOF analysis. Simple models specified in MDOF analysis, such as mass proportional or initial-stiffness proportional damping matrix, and more complicated models, such as the methods of Caughey [1960] and Wilson and Penzien [1972], will lead to a constant damping matrix, and the ductility dependence of their behaviour will also be qualitatively described by the SDOF results in this section. For SDOF systems, the constant value of the damping coefficient is determined with respect to the initial vibration frequency, ω i = k i / m, and a specified fraction of critical damping, ξ: c = 2 mω ξ = 2ξ. (8) i mk i In the substitute structure model, the damping coefficient is determined with respect to the secant stiffness, k eff, and will therefore be given by: c 2ξ eq, v = mk. (9) If the velocity is the same for the inelastic model and the substitute structure, then the viscous energy dissipation will be identical if: where, v λ 1 eff ξeq = ξ, (10) ki µ λ1 = =, (11) k 1+ rµ r eff and µ and r are the displacement ductility and post-yield stiffness ratio, respectively. The first equality in Eq. (11) applies for any hysteresis model, whereas the second

12 12 M. J. N. Priestley & D. N. Grant equality only applies for models with a bilinear backbone, such as the Bilinear, modified Takeda, and Flag models considered in previous sections. For small r, we have λ1 µ. Equation (11), which is plotted in Fig. 12(a), implies that if a constant damping coefficient equivalent to 5% of critical damping is assumed to be appropriate, then for an expected displacement ductility of µ = 4, and post-yield stiffness ratio r = 0.05, an equivalent damping of ξ eq,v = 9.33% should be assumed in DDBD Tangent-stiffness proportional damping coefficient When the damping coefficient is proportional to the instantaneous value of the stiffness, it is updated whenever the stiffness changes. At any instant, c is given by: kt 2ξ c = 2 mωiξ = kt. (12) k ω Using the approach described above, the following expression for the fraction of critical damping required for the substitute structure may be obtained:, v = ξ λ1 λ2 where λ 1 is obtained from Eq. (11), and λ 2 is given by: i i ξeq, (13) ( µ 2) 1 r 1 µ 2 2 µ 1 λ2 = r + cos, (14) 2 π µ µ for Bilinear hysteresis, and a more complicated expression for the modified Takeda model, given in full in Ref. [Grant et al., 2004]. Note that the r term in Eq. (14) represents the dissipated energy for a linear viscoelastic system with stiffness r ki. In Fig. 11, this corresponds to the area of the smaller ellipse partly traced out by the damping force between points Y and U; the extra spikes in the damping force from U to Y are given by the second addend in Eq. (14). The inverse cosine is a result of the assumption of harmonic steady-state response in the nonlinear model. Equation (14), for Bilinear hysteresis, and the corresponding expression for modified Takeda hysteresis, are plotted in Figs. 12(b) and 12(c), respectively Rayleigh damping model Rayleigh damping is commonly used in MDOF, nonlinear analyses, where modal analysis is not possible. The damping matrix is specified as a sum of mass and stiffness proportional terms, with two constants to be specified in terms of desired modal damping ratios. In SDOF analysis, where only one damping ratio can be specified, the Rayleigh damping assumption is over-defined, and an additional assumption about the proportions of mass and stiffness damping must be made. The results from such an assumption, properly calibrated for typical structures, can be used to give insight into the nature of viscous damping in MDOF analyses using Rayleigh damping. The damping coefficient for tangent-stiffness Rayleigh damping for a SDOF system is given by:

13 Equivalent Viscous Damping in Seismic Design and Analysis 13 c = α m + β, (15) where α and β are the same coefficients introduced in Sec. 2. In this case, the resulting damping model is a mixture of the models considered in Secs and 4.1.2, and the correction factors, λ 1 and λ 2 must be proportioned accordingly. For initial-stiffness proportional Rayleigh damping, the damping coefficient is constant, and the results from Sec may be used directly. The factor λ 1, defined by Eq. (11) is a function of the effective stiffness of the system, and therefore does not change with damping model. The factor λ 2, given in Eq. (14) for Bilinear hysteresis, is a function of the ratio of damping coefficients for instantaneous and initial system properties. For this reason, for tangent-stiffness proportional Rayleigh damping, the post-yield stiffness ratio must be replaced with the ratio of damping coefficients, which for the Bilinear model is given by: r * k t α m + β r ki =. (16) α m + β k As an example, if the mass and stiffness terms initially contribute in the ratio 5:1, as calculated in Sec. 2 for typical MDOF structural properties, then: r i 5 + =. (17) 6 * r For r / 6 0, r * 5/ 6, which is equal to the assumed fraction of mass proportional damping. For the modified Takeda model, the effective post-yield stiffness ratio must be evaluated based on the ratio of reloading stiffness under steady-state response to the unloading stiffness, and will be ductility-dependent. The effective post-yield stiffness ratio given by Eq. (17) is clearly greater than r for 0 r 1, and therefore λ 2 is closer to one, and the total multiplier λ 1 λ 2 is greater, than that obtained for pure tangent-stiffness proportional damping. In MDOF applications, lower frequencies will have a larger contribution of mass, and therefore λ 1 λ 2 will be larger than for higher frequencies, as discussed in Sec Numerical method for viscous damping in DDBD The analytical expressions derived in the previous section assume that the viscous energy dissipation in a single cycle of steady-state harmonic response is the fundamental measure of elastic viscous damping. As discussed in the previous sections, this is the same approach taken for the hysteretic component of the equivalent viscous damping in Jacobsen s approach [Jacobsen, 1930; 1960]. The factor λ 1 derived in Sec is an exact correlation between the damping coefficients of the nonlinear model and the substitute structure, as the viscous damping remains linear in both cases. The λ 1 factor adjusts the fraction of critical damping specified in terms of initial stiffness to one appropriate for secant stiffness, which, provided the velocity history is the same in both cases, results in exactly the same

14 14 M. J. N. Priestley & D. N. Grant damping force for the duration of loading. For this reason, Eq. (11) does not require numerical validation. The derivation of factor λ 2 for tangent-stiffness proportional damping and Rayleigh damping, however, clearly involves some degree of approximation. It is not immediately obvious what effect replacing the nonlinear viscous damping for Bilinear hysteresis (the solid lines in Fig. 11(b)) with an equivalent linear viscous damping (dashed line, scaled by λ 2 ) will have on the response of a SDOF system. For the modified Takeda hysteresis model, the viscous damping force is even more complicated, particularly when the loading is non-harmonic, as different branches of the hysteresis model exhibit significantly different stiffnesses. At any instant in time, the actual damping force will be different in the nonlinear and substitute structure models, and what is important is the effect of the viscous energy dissipation on the maximum displacement response. It is difficult to use a direct comparison of the response of a nonlinear system with that of the corresponding substitute structure to determine the correction factor, λ 2. For the former system, the effect of damping due to hysteresis cannot be separated from that due to viscous dissipation, and the substitute structure model will require an approximation of the equivalent viscous damping due to both components. Conversely, existing studies of the hysteretic component of equivalent viscous damping [e.g. Judi et al., 2000; Miranda and Ruiz-Garcia, 2002] that do not consider the model- and ductilitydependence of the elastic viscous component do not properly separate the effects of the two halves of Eq. (1). In that case, adopting an elastic viscous damping equal to zero in both DDBD and ITHA, (as done by Blandon et al. [2005]), could isolate the effects of hysteresis. We cannot achieve the converse by setting the hysteretic dissipation to zero as we would no longer have a nonlinear system. For this reason, the system described in Fig. 13 has been used to study the effects of tangent-stiffness proportional and Rayleigh damping on the response of a SDOF structure. In this system, the restoring force, F s, is obtained from a linear elastic model with stiffness equal to the effective stiffness of the substitute structure model (Fig. 13(a)). The calculation of the damping force (Fig. 13(b)), however, assumes that the stiffness is actually equal to the tangent stiffness of a simulated hysteretic model, k t,sim (simulated Bilinear hysteresis is shown in the dotted line in Fig. 13(a)). The damping level of the simulated nonlinear system, ξ, is adjusted until the peak displacement response under a given input ground motion matches that of a substitute structure with specified damping, ξ eq,v. The ratio ξeq, v / λ1 ξ is then equal to λ 2, from Eq. (13). Note that hysteretic damping (or its linearised equivalent) is not included in either the substitute structure or the simulated nonlinear structure. The procedure described above was carried out for target ductility values between 1.0 and 10.0, over a suite of 40 ground motions records from the FEMA/SAC database [Woodward-Clyde Federal Services, 1997]. The records, from the Los Angeles (LA) and near fault (NF) subsets of the database, were scaled to obtain the target displacement in the substitute structure, and therefore the actual hazard level to which the ground motions correspond is irrelevant, provided that the duration of each record is somewhat consistent

15 Equivalent Viscous Damping in Seismic Design and Analysis 15 with its intensity. Although the analytical relationships derived in Sec. 4.1 do not predict period-dependence, the simulations were carried out for SDOF initial stiffness corresponding to effective periods of 1.0 sec and 2.0 sec at a ductility of 4. Finally, the post-yield stiffness ratio and, for the modified Takeda model, the unloading parameter, γ, (see Eq. (6)) were varied, although only the results for γ = 0. 5 are reported here. The median results across the ground motion suite for converged values of λ 1λ2 are shown in Fig. 14. The total multiplier is plotted, rather than just the unknown value λ 2, as it is the product that is required for DDBD applications, and this allows simplified design equations to be obtained directly from Fig. 14 (see below). Figures 14(a) and 14(b), for Bilinear and modified Takeda hysteresis, respectively, show that the analytical expressions developed in Sec. 4.1 generally underestimate the correction factors obtained from the numerical simulations. For both models, it may be observed that the perioddependence is also negligible. Increasing the post-yield stiffness also tends to increase the factor λ 1 λ 2 towards unity, although for the range of r values appropriate for modelling reinforced concrete elements with the modified Takeda model, the dependence is small. Obviously, for the Bilinear model, as r tends to one, the response tends to linear, and λ 1λ2 1. For the modified Takeda model, this limit does not hold, as the applicability of the model is restricted to values of r less than or equal to ( 1 µ γ 1) /( µ 1) [Grant et al., 2004]. Although not shown here, the dependence of the numerical results on the Takeda unloading parameter γ is also negligible [Grant et al., 2004]. Although just the median results are shown in the figures, it should be noted that there was a relatively large amount of scatter in the results for all the ground motions in the suite. It is particularly notable that, in rare cases, the simulated tangent stiffness model with ξ = 0% resulted in smaller displacements than in the substitute structure model with ξ eq, v = 5%. This somewhat counterintuitive result shows that, in some cases, a system with some viscous damping has a larger peak response than one with none. Clearly, the peak response cycle is also affected by the maximum displacement of the prior cycle, which cannot be described by an equivalent linearisation based on steadystate energy dissipation. In any case, these isolated cases do not affect the results shown in the figures, as the median values are not affected by such outliers. The following equations, linear in µ for a given value of r, provide a good match of the median values from the numerical simulations, for realistically attainable ductility levels, 1 µ 6 : Bilinear model: λ = ( 1)( 1 r) λ ; (18a) 1 2 µ Takeda model: λ = ( 1)( 1 r) λ. (18b) 1 2 µ Equations (18a) and (18b) should only be used for the range of r values used in the numerical analyses 0 to 0.2 for the Bilinear model, and 0 to 0.05 for the modified Takeda model although at least Eq. (18a) exhibits the correct limit as r tends to 1.0. Using the same example as considered in Sec , the required damping to be used in DDBD to model 5% tangent-stiffness damping when µ = 4 and r = 0.05 is, from Eq.(18b), ξ eq,v = 3.65%.

16 16 M. J. N. Priestley & D. N. Grant The relatively poor fit of the analytical expressions observed in the numerical simulation process shows that the problem of obtaining the correct level of equivalent viscous damping is not as simple as assumed in the derivations in Secs and For this reason, we believe that Eqs. (18a) and (18b) are more appropriate for design than the analytical expressions in Eq. (14) and Ref. [Grant et al.,2004], for Bilinear and modified Takeda hysteresis, respectively. Because the simplified design equations combine λ 1 and λ 2 for tangentstiffness proportional damping, for Rayleigh damping it would be necessary to divide * Eq. (18) by Eq. (11), and to replace r by r from Eq. (16). Alternatively, simplified expressions could be developed directly for λ 2, as in Ref. [Grant et al., 2004]. This has not been presented here, as we believe that tangent-stiffness proportional damping is the best model for real non-hysteretic energy dissipation, and therefore Eqs. (18a) and (18b) should be used in the design of real buildings. A second numerical approach can also be carried out to obtain expressions for λ 1 λ 2. Comparing the inner and outer solid lines in Fig. 11(b), it is apparent that the only difference between the constant damping and tangent-stiffness proportional models is in the viscous damping force, provided both analyses reach the same ductility level. Therefore, if the initial- stiffness proportional model is run with a certain input motion and damping level, ξ, then the tangent stiffness model should reach the same ductility for a damping level ofξ / λ2. Actually, this is an oversimplification, as hysteresis and energy dissipation under smaller hysteretic cycles will be different in each case, but this simplification is consistent with the design procedure in which only peak values of displacement are considered. The results from this numerical procedure are not shown here, but they are very similar to the results obtained using the simulated tangent stiffness approach. 5. Conclusions It is reasoned that elastic viscous damping in structures is more realistically modelled in inelastic time-history analyses by a tangent-stiffness proportional viscous damping coefficient than by a constant damping coefficient. This is in contradiction to common analytical practice. Comparative inelastic analyses of single-degree-of-freedom systems with different hysteretic rules, and either a constant damping coefficient or tangent-stiffness proportional elastic damping showed that the choice of damping model can give a very significant influence on peak response displacements, with the tangent-stiffness assumption invariably resulting in larger displacements. The difference increases as displacement ductility increases, and is dependent on initial period and hysteresis rule. It was found that for the reported analyses, the equal-displacement approximation for moderate and long period structures, though reasonable when a constant damping coefficient was assumed, was generally significantly non-conservative when tangentstiffness proportional damping was adopted for the time-history analyses. It is shown that as a consequence of the difference between the initial stiffness and the effective stiffness used to characterise response in Direct Displacement-Based

17 Equivalent Viscous Damping in Seismic Design and Analysis 17 Design, it is necessary to modify the value of elastic viscous damping from the value referenced to the initial stiffness. The necessary correction factor is more significant if the elastic viscous damping coefficient is assumed to be constant, as assumed in most studies of force reduction factors or equivalent linearisation of structural response. In verification of designs developed by DDBD, it is essential that the assumptions for elastic damping made in the design and time-history analysis are compatible. References Blandon, C. A., and Priestley, M. J. N. [2005] Equivalent viscous damping equations for direct displacement-based design J.Earthquake Engg. 9(Special Issue 1), xxxx Carr, A. J. [2001] Ruaumoko a program for inelastic time-history analysis Dept.C.Engg. Univ. Canterbury, New Zealand. Caughey, T. K. [1960] Classical normal modes in damped linear systems, J. Appl. Mech. 27, Grant, D. N., Blandon, C. A. and Priestley, M. J. N. [2004] Modelling inelastic response in direct displacement-based design Rept. No. ROSE 2004/02,European School for Advanced Studies in Reduction of Seismic Risk, Pavia xxxpp Jacobsen, L. S. [1930] Steady forced vibrations as influenced by damping, Transactions ASME 52, Jacobsen, L. S. [1960] Damping in composite structures, in Proceedings of the 2nd World Conference on Earthquake Engineering, volume 2, Tokyo and Kyoto, Japan, pp Iwan, W. D. [1980] Estimating inelastic response spectra from elastic spectra Earthquake Engg. And Struct. Dynamics 8, Miranda, E. [2000] Inelastic displacement ratios for structures on firm sites J.Struct.Engg. 126, Miranda, E and Ruiz-Garcia, J. [2002] Evaluation of approximate methods to estimate maximum inelastic displacement demands Earthquake Engg. And Struct. Dynamics 31, Newmark, N. M. and Hall., W. J. [1982] Earthquake Spectra and Design EERC Berkeley. Otani, S [1981] Hysteretic models of reinforced concrete for earthquake response analysis J.,Faculty of Engg. Univ. Tokyo, XXXVI(2) 24pp. Otani, S. and Sozen, M. A. [1972] Behavior of multistory reinforced concrete frames during earthquakes, Structural Research Series No. 392, Civil Engineering Studies, University of Illinois, Urbana. Priestley, M. J. N. [2000] Performance-based seismic design Keynote address, 12-WCEE, Auckland, 22pp. Stanton, J et al [1998] PCI Journal (to be added) Veletsos A. S. and Newmark, N. M. [1960] Effect of inelastic behaviour on the reponse of simple systems to earthquake motions Proc. World Conference on Earthquake Engineering, Japan, V.2, Wilson, E. L. and Penzien, J. [1972] Evaluation of orthogonal damping matrices, Int. J. Numerical Methods in Engg. 4, Woodward-Clyde Federal Services [1997] Suites of ground motion for analysis of steel moment frame structures. Report No. SAC/BD-97/03, SAC Steel Project.