To better observe the effects of SSI on the response of the piers, the mean values of DDR and TDR are plotted together in Figure 7.5. This figure demonstrates that SSI decreases ductility demands at the expense of increasing total displacement demands, although it is emphasized again that relying on the mean values as presented in Figure 7.5 can be misleading in regard to the effects of SSI.
2.0 1
1.8 qe sẽ #e rẽ me re nem inl Ductility Demand Ratio (DDR)
A Total Displacement Ratio (TDR) |
T
124--- 1.0 4:
08 +. |
0.0
Mean Response Ratio
1 { ' Ị 1 i 1 É- 1 T | | ‡
0.0 0.5 1.0 1.5 2.0 2.5
Fixed-Base Period (s)
Figure 7.5: Mean response ratios (DDR and TDR)
It is also informative to relate DDR and TDR to an overall system parameter rather than merely the fixed-base period of piers. An important system parameter in regard to the effects of SSI is the stiffness of the pier relative to the stiffness of the foundation, described by the pier-to-foundation stiffness ratio. As was observed in Chapter 2, and has been shown by other researchers (e.g. Finn 2004b), the period elongation of the system due to SSI is correlated to the pier-to-foundation stiffness ratio. Therefore, the ratio of the system initial period to the period of the fixed-base pier is a convenient system parameter
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that can be employed. Greafter system-to-fixed-base period ratio (T;yz/T) Indicates greater period elongation and more flexibility at the base of the pier due to higher pier-to- foundation stiffness ratio. Since piers with shorter periods are stiffer, the period elongation increases with decreasing natural period of the fixed-base pier (see Chapter 2).
Obviously one can expect to observe greater effectiveness of SSI with greater flexibility of the base, i.e. with greater period elongation.
Figure 7.6 depicts the plots of the mean DDR and the mean TDR versus the dimensionless period ratio, Ts,./T. An interesting observation in Figure 7.6 is that the relationship between the demand ratios and the period ratio Tsy./T of the system is rather linear (although it should be recognized that the selection of the prototype systems based on the piers fixed-base natural period has resulted in the lack of data points between Tsy/T of 1.5 and 3.0). An advantage of the presentation of results as a function of Tsyx/T is that the results obtained from the analysis of the prototype systems of this study can be compared to any similar study with different prototype systems but with similar Ts,./T ratios. If the comparisons show that DDR and TDR thus obtained can be applicable to different soil-foundation-structure systems based on their Tsy./T value, then DDR and TDR curves can be used to modify the response obtained without SSI to estimate the demands including SSI. It is reminded, however, that using the mean values of DDR and TDR without consideration of the scatter of results may obscure some aspects of the system response. Therefore, a probabilistic approach is required so that DDR and TDR can be reliably used to estimate the effects of SSI on the response of bridge piers. The probabilistic assessment of the effects of SSI and further discussion on the application of DDR and TDR for performance-based estimation of the effects of SSI is presented in Chapter 8.
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2.0 T T
1.8 +4 @ Ductility Demand Ratio(DDR) |----1---
1677 # Total Displacement Ratio (TDR) i--~-a7--77--7 77-7 1.4 ơ
1.2 + 1.0 +~
0.8 4 0.6 5 0.4 0.2 + 0.0
Mean Response Ratio
1.00 1.10 1.20 1.30 1.40
Figure 7.6: Mean demand ratios (DDR and TDR) as functions of the period ratio Tsys/T
7.4 Summary
Effects of SSI on the behaviour of the system were explored by presenting the pier ductility demand and the pier total displacement ratios (DDR and TDR). It was observed that:
e The effects of SSI on the ductility and the total displacement demands is increased with decreasing period of the piers, or with increasing period ratio of the system (Tsys/T).
e In most cases, SSI reduces the ductility demand of the piers (DDR < 1.0), however, this is not true for all cases and increased ductility demands were also observed. It is more likely that piers with longer periods (lower T,,;/T) experience greater ductility demands due to SSI
e The reduction of the ductility demands is at the expense of increasing total displacement demands.
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e Relying on the mean values of DDR and TDR can be misleading in regard to the effects of SSI on the seismic demands of the system. Proper observation of the data requires accounting for its dispersion. Therefore, results obtained should be studied probabilistically so that accurate conclusions in regard to the effects of SSI could be drawn.
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8 PROBABILISTIC ASSESSMENT OF THE EFFECTS OF SOIL-STRUCTURE INTERACTION
It was explained in Chapter 7 that due to the dispersions of DDR and TDR, relying merely on their mean values can be misleading in regard to estimating the effects of SSI on the ductility and total displacement demands of the bridge piers. Therefore, results obtained should be studied probabilistically to properly account for the uncertainties of DDR and TDR, so that they can be used for seismic demand estimations within the framework of performance-based design. The source of uncertainties for DDR and TDR are the input earthquake ground motions used in seismic demand estimations of the bridge piers. The input motions are random in nature and hence are DDR and TDR.
Besides the input motions, the system parameters also have uncertain natures and should be treated as random variables. Therefore, proper probabilistic assessment of DDR and TDR should also account for the randomness of the system parameters to which DDR and TDR are dependent. In this regard, one of the objectives of the probabilistic assessments presented in this chapter is to account for the uncertainties of DDR and TDR by estimating the probabilities that they are greater than predetermined target values.
Target values are chosen based on the performance objectives. For instance, the probability of DDR>1.0 defines the probability of SSI increasing the ductility demand of the pier, which is the probability of ignoring SSI being an unconservative assumption.
Moreover, with the evolution of performance-base design methodologies, probabilistic estimate of the performance of structures under earthquake loadings is needed to be accounted for explicitly as an integrated part of structural design so that the system can be designed and optimized to meet various limit states with different reliabilities tailored
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to the needs of the project. Hence, another objective of the probabilistic assessments presented in this chapter is to estimate the effects of SSI on the performance of the piers with given target reliabilities. For instance, TDR can be used to estimate the total displacement of a bridge pier based on the total displacement of that pier without SSI.
This can be done by multiplying the total displacement of the fixed-base pier by TDR.
The mean value of TDR can be used for this purpose; however, there would be about 50% chance that TDR is greater than the mean TDR which may represent too much uncertainty. To find TDR with higher level of confidence, it can be estimated for a target reliability that reflects the corresponding desired level of confidence. For instance, ifr is found such that the probability of TDR<r is p, then TDR=r can be used to estimate the effects of SSI on the total displacement of the pier with the reliability of p. Note that the reliability analysis is performed by estimating the probability of not meeting the performance objective, i.e. the probability of TDR>r (in other words, if the probability of TDR>r is P, then P is the probability that total displacement is greater than that predicted by TDR).
In the following sections, first the methodology used for reliability analysis is explained.
Then results of the reliability analyses are presented towards the stated objectives of this chapter, i.e. probabilistic data processing, and application for performance design.
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