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EFFECTS OF DAMPING MODELLING ON RESULTS OF TIME-HISTORY ANALYSIS OF RC BRIDGES

NIGEL PRIESTLEY Centre of Research and Graduate Studies in Earthquake Engineering and Engineering Seismology (Rose School), Istituto Universitario di Studi Superiori (IUSS) Pavia, Italy

MICHELE CALVI Department of Structural Mechanics, Univerita degli Studi di Pavia, Pavia Italy

LORENZA PETRINI Department of Structural Engineering, Politecnico di Milano, Italy

CLAUDIO MAGGI Department of Structural Mechanics, Universita degli Studi di Pavia Italy

Abstract: The choice between initial-stiffness and tangent-stiffness elastic damping for modelling dynamic response of reinforced concrete bridge structures is discussed. It is argued that initial-stiffness elastic damping, which has been extensively used in most previous analyses, is inappropriate, and results in large, spurious damping forces in the inelastic range of response. When tangent-stiffness elastic damping is used, damping forces are reduced, and as a consequence displacements are increased to the extent that the “equal-displacement” approximation is invalid for most hysteresis rules. Results of a simple static/dynamic experiment to investigate which damping model is most appropriate conclusively demonstrate that initial-stiffness elastic damping is non-conservative

Key words: damping, inelastic response, time-history analysis, dynamic test, bridge pier

Nigel Priestley, Michele Calvi, Lorenza Petrini, Claudio Maggi

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1. INTRODUCTION

It is common to specify a level of elastic damping in inelastic time-history analysis (ITHA) to represent damping in the initial stages of response, before hysteretic damping is activated. This is normally specified as a percentage, typically 5%, of critical damping. There are a number of ways this damping could be defined, but the principal difference is whether the damping force is related to the initial or tangent stiffness.

Typically, research papers reporting results on single-degree-of-freedom (SDOF) ITHA state that 5% elastic damping was used, without clarifying whether this has been related to the initial or tangent stiffness. With multi-degree-of-freedom (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 our understanding that many analysts consider the choice of the 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 and tangent-stiffness proportional damping matrix could be significant, particularly for short-period structures.

With initial stiffness elastic damping, the damping coefficient is constant throughout the analysis, even in the inelastic range of response, and is based on the initial elastic stiffness. With tangent-stiffness damping the damping coefficient is proportional to the instantaneous value of the stiffness and it is updated whenever the stiffness changes. With the case of elasto-plastic response, the tangent-stiffness damping force will be zero while the structure deforms along a yield plateau.

There are three main reasons for incorporating 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 elastic 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.

• Non-structural damping: Hysteretic response of non-structural elements, and relative movement between structural and non-structural elements may result in an effective additional damping force.

Discussing these reasons in turn, it is noted that hysteretic rules are generally calibrated to experimental structural data in the inelastic phase of response. Therefore additional

1st US-Italy Seismic Bridge Workshop 3

elastic damping should not be used in the post-yield state to represent structural response except when the structure is unloading and reloading elastically. If the hysteretic rule models the elastic range nonlinearly (as is the case for fibre-element modelling) 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 a linear representation of the elastic range, and which 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 in the structural post-yield stage, 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 shown elsewhere (Priestley et al (2007)) that non-structural damping in modern buildings is likely to contribute no more than 1% effective damping in the inelastic range of response. With bridges it is hard to see where non-structural damping could originate. It is instructive in determining the influence of alternative elastic damping models to consider the steady-state, harmonic response of an inelastic SDOF oscillator subjected to constant sinusoidal excitation. This enables direct comparison between hysteretic and elastic damping energy, and also between elastic damping energy using a constant damping coefficient and tangent-stiffness proportional damping models. To this end, Fig.1 shows response of a simple SDOF oscillator with initial period of 0.5 sec and a “thin” modified Takeda hysteresis rule subjected to 10 seconds of a 1.0 Hz forcing function. 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.1(a) (initial-stiffness proportional damping) and Fig.1(b) (tangent-stiffness proportional damping). In each case the hysteretic response associated with nonlinear structural response is plotted on the left, and the elastic damping force-displacement 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


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hysteretic damping is equivalent to about 20% of critical damping. This apparent anomaly is due to the different reference stiffness used. The elastic damping is related to the initial stiffness, whereas the hysteretic damping is related to the secant stiffness to maximum response.

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(a) Analysis with Initial Stiffness Damping

(b) Analysis with Tangent Stiffness Damping Fig.1 Steady-state Inelastic response of an SDOF oscillator with “Thin” Modified Takeda hysteresis

When the elastic damping is tangent-stiffness proportional, as we believe to be most appropriate for structural response, the elastic damping energy is greatly reduced, as can


be seen by comparing the upper and lower right-hand plots of Fig. 1. 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. Analyses of SDOF systems subjected to real earthquake records (Priestley and Grant (2005)) show that the significance of the elastic damping model is not just limited to steady-state response. Figure 2 shows a typical comparison of the displacement response for SDOF oscillators with initial stiffness and tangent stiffness elastic damping. In this example the El Centro 1940 NS record (amplitude scaled by 1.5) has been used, the initial period was 0.5 seconds, a Takeda hysteretic rule with second slope stiffness of 5% was adopted, and the force-reduction factor was approximately 4. The peak displacement for the tangent- stiffness elastic damping case is 44% larger than for the initial-stiffness damping case, indicating a very significant influence.

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A selection of results from a series of analyses of oscillators with different initial periods between 0.25 and 2.0 seconds subjected to a suite of ATC32 (1996) specrrum compatible accelerograms is shown in Fig.3, which presents the ratio of peak inelastic displacement response to peak elastic displacement response for three different hysteresis rules, and three different force-reduction factors, applied to the average elastic peak response force.


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Fig.3 Response of SDOF Oscillators to ATC32 Spectrum-Compatible Accelerograms


From examination of Figure 3 it will be noted that there is a significant difference between the response of the initial-stiffness and tangent-stiffness (identified as IS and TS respectively) models, that this difference is rather independent of initial period, for T>0.5 seconds, that the difference increases with force-reduction factor, and is dependent on the hysteretic rule assumed. It will also be noted that though the “equal displacement” approximation (represented by a displacement ratio of 1.0 in Figure 3) is reasonable for initial stiffness damping and initial periods greater than T = 1.0 seconds, it is significantly non-conservative for tangent-stiffness elastic damping.

2. TESTS ON BRIDGE PIERS

In order to investigate the validity of the arguments supporting tangent-stiffness initial damping, made above, a simple experimental program was undertaken at the EUCentre, Pavia. Two identical SDOF hollow-section cantilever bridge piers were constructed, based on a model/prototype scale of 1:4. Details of reinforcement and general dimensions are shown in Fig.4.

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Figure 4 Dimensions of Piers for Static and

The first test was carried out under static cyclic displalevels of displacement ductility of µ = 1, 2, 4 and 6, wlevel, followed by an elastic cycle to enable stiffness degra

f’c = 39MPa fy = 514 MPa Mass = 7.8 tonnes

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cement-controlled excitation at ith three cycles at each ductility dation to be determined.


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Predicted response was obtained using SeismoStruct (SeismoSoft (2007)), a fibre-based multi-purpose structural analysis package. The experimental response was also used to calibrate the parameters of a modified Takeda hysteresis rule. Experimental force-displacement hysteresis response is compared with predictions by SeismoStruct and the Takeda modelling in Figs 5 and 6 respectively.

It will be noted that there is a small, but significant apparent strength eccentricity in the experimental results, with higher values obtained in the positive force-displacement quadrant than in the negative quadrant. This is partially a consequence of the expected higher flexural strength in the initial loading direction, but is also probably due in part to the reinforcement cage being slightly eccentric in the column diameter.

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Figure 5 SeismoStruct Simulation of Static Test

It will also be noted that the SeismoStruct simulation slightly underestimates the flexural strength, and results in hysteresis loops that absorb more energy than the experimental loops. This is a consequence of the difficulty in modelling shear deformation in fibre-element based analyses. Strength degradation on cycling to a specified ductility level is also slightly under-predicted by the SeismoStruct modelling, but overall, the simulation is excellent.

The Takeda hysteresis modelling was, as noted above, adjusted to provide the best fit to the experimental results. Strength degradation was not included in the modelling, but


otherwise the agreement with experimental results is excellent.

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Figure 6 Takeda Hysteresis Simulation of Static Test

The second unit was tested on a shake table using an accelerograms based on the 1984 Morgan Hill record. This record was chosen based on preliminary analyses with a number of accelerograms where the aim was to provide the largest difference between predicted peak displacement using initial-stiffness and tangent-stiffness elastic damping models. The record was scaled in the time-domain as a consequence of the model scale, and also scaled in amplitude to produce a predicted peak displacement corresponding to displacement ductility demand of µ = 6, based on the tangent-stiffness prediction. This represented the maximum stable displacement response of the static model (see Figs 5 and 6) – at larger displacements strength degradation was rapid. It should be noted that the peak predicted displacement response using initial-stiffness damping and the Takeda simulation was of the order of µ = 3. The difference was larger when damping was added to the SeismoStruct simulation, with initial-stiffness damping predicting average peak displacements (average of positive and negative peaks) only 39% of the tangent stiffness predictions. It was thus felt that the dynamic test could be confidently expected to establish which damping model was more appropriate for this test.

The recorded table excitation was very close to the modified Morgan Hill record, though peak acceleration amplitudes were about 10% higher than expected. As a consequence, predictions from the SeismoStruct and Takeda simulations were re-run using the


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recorded table accelerations. Comparisons of recorded experimental displacement response and predictions from the two analysis methods for the first 12 seconds of response are included in Fig. 7.

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(b) Takeda Simulation Figure 7 Comparison of Experiment and Predicted Displacement Time-Histories for Dynamic Test

Considering first the experimental response, it is seen that the peak displacement rises by T = 3.5 sec. in three cycles to about 0.1m, the value corresponding to µ = 6 from the


static test, and hence the largest stable static response level. Following this the experimental response indicates a significant residual displacement (about 0.05m) with another large displacement pulse between T = 10 and 10.5 seconds. The negative displacement in this pulse was 0.15m, corresponding to a displacement ductility factor of µ = 9, and was accompanied by severe strength degradation. This was eventually accompanied by fracture of all flexural reinforcement at the column base, and catastrophic failure of the column.

The SeismoStruct prediction of response (Fig.7a) based on a tangent-stiffness elastic damping model provides a good representation of the experimental response, particularly up to T = 5.5 seconds. After this the residual displacement is slightly underestimated. Beyond T = 10 seconds, the strength degradation resulting from fracture of the flexural reinforcement was not captured in the SeismoStruct simulation, and displacements were underestimated. Considering peak displacements up to T = 10.2 seconds, the SeismoStruct peaks were 76% and 93% of the experimental peak positive and negative displacements respectively. On the other hand, displacements predicted by the initial-stiffness elastic damping simulation severly underestimated the peak experimental displacements, being 27.5% and 36% of the peak positive and negative displacements.

The Takeda line-element simulations were carried out using Ruaumoko (Carr, (2005)). It will be seen from Fig.7b that the tangent-stiffness simulation provided extremely close estimates of the experimental response in the initial cycles up to the peak static displacement capacity of 0.1m, which occurred at about T = 3.5 seconds, while the initial-stiffness simulation underestimated the displacements by about 30% on average, except in the initial elastic cycles (µ<1). Beyond T = 3.5 seconds the residual displacement was underestimated by both initial-stiffness and tangent-stiffness models, but the peak displacement at T = 10.2 seconds was again captured well by the tangent-stiffness model. Considering peak displacements up to T = 10.2 seconds, the tangent-stiffness model peaks were 104% and 102% of the experimental positive and negative displacement peaks, while the initial-stiffness values were 55% and 67% of experimental peaks respectively.

3. DISCUSSION OF RESULTS

The different analytical results provide compelling support for the use of tangent-stiffness, rather than initial-stiffness elastic damping. It should be noted, however, that for the fibre-element based SeismoStruct simulation, it can be argued that no elastic damping should be included, as the nonlinearity in the “elastic” range of response should be directly modelled. A further analysis was thus carried out with SeismoStruct, with no added elastic damping. The results, which are not included in Fig.7a), were very similar, though slightly larger, than those obtained with tangent-stiffness damping. Peak


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displacements were increased to 91% and 98% of the experimental peak positive and negative displacements.

For comparison, the Takeda analysis was also rerun for zero elastic damping. Displacement peaks increased to 118% and 106% of experimental peaks.

It will be noted that the predictions for 5% initial-stiffness damping from SeismoStruct and Ruaumoko differ significantly (compare Figs 7a and 7b). This is because the SeismoStruct simulation used the initial un-cracked section stiffness, whereas the Ruaumoko simulation was based on effective stiffness to first yield. A stiffness difference by a factor of approximately 2.5 thus resulted between the two simulations, resulting in a much higher elastic damping force for the SeismoStruct analyses. The question could thus be asked to advocates of initial-stiffness damping: what value of initial stiffness should be used.

4. CONCLUSIONS

Common sense considerations of structural behaviour indicate that initial-stiffness based elastic damping is inappropriate for dynamic analysis, resulting in elastic damping forces that are unrealistically high. Tangent-stiffness elastic damping appears to be more appropriate, and results in increased displacements compared with initial-stiffness predictions, particularly at high ductilities, and when hysteretic energy is low.

Results of a shake-table test of a simple bridge pier confirmed that tangent-stiffness elastic damping gave the best prediction of displacement response when a simple Takeda simulation, with linear response in the elastic range was used. When a more sophisticated fibre-element simulation was used, the best simulation was obtained when no elastic damping was used.

5. REFERENCES

Applied Technology Council (1996) “ATC32: Improved Seismic Design Criteria for California Bridges – Provisional Recommendations” Redwood City, CA, 256pp

Carr, A.J. (2005) “Ruaumoko User Manual” Dept of Civil Engineering, University of Canterbury. Otani, S. (1981) “Hysteretic Models of Reinforced Concrete for Earthquake Response Analysis”

Journal of Faculty of Engineering, University of Tokyo, Vol.36(2) pp 407-441 Priestley, M.J.N. and Grant, D.N. (2005)“Viscous Damping in Seismic Design and Analysis”

Journal of Earthquake Engineering Vol.9 (SP2) pp 229-255 Priestley, M.J.N., Calvi, G.M., and Kowalsky, M.J. (2007) “Displacement-Based Seismic Design of

Structures” IUSS Press, Pavia, 721 pp. SeismoSoft (2007) “SeismoStruct – A computer program for static and dynamic nonlinear analysis

of framed structures” available from URL:http://www.seismosoft.com

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