Results presented in the previous section were concerned with the probability of SSI increasing the demands of the piers regardless of the magnitude of the amplification of demands. Moreover, probabilities can be estimated for SSI decreasing the demand and not just increasing the demand. Hence, reliability analyses can be carried out for various performance objectives, defined by the values chosen for Djim as discussed in Section 8.1.
The curves of Figures 8.9 and 8.10 were obtained for Dii,= r, where r is from 0.8 to 1.5, and show the probability of DDR>r and TDR>r, respectively, as functions of Ts,,/T. It can be observed, for instance in Figure 8.9, that for a pier with Tsys/T of 1.05, while the probability of DDR>1.0 is about 30%, it is less than 5% for DDR>1.3 which is a much lower probability. Therefore, if the column has 30% ductility reserve, then one can say that ignoring SSI will not pose a significant risk to the pier (less than 5%). As another example, for a system with Tsy./T of 1.20, the probability of DDR>0.9 is less than 10%, which means that the ductility demand can be reduced by 10% with more than 90%
confidence. Such reduction of demand can become a source of saving especially in the retrofit of bridge piers. It is reminded that the probabilities here are not total probabilities and they are conditional on the occurrence of all the ground motions used in this study.
Calculation of total probabilities requires estimation of the probability of occurrence of the input ground motions which is not within the scope of this work.
150
1.4
r=0.8 os we ep =09
msmmmasae TS |, - |~]
—[= l@ẽ
T ' ù ' ì :
Tsys/T
0.5
S—:ctce tin wr | eh se tee ee Poo NL AAA OO ' Reto not ti wt wa Ị t RRR REE ; ' i a 8 ft 1 ' Ễ : { t ' : ao & : i i ' t ‡ i ‘ ‘ a bk t ' : : Ễ ..ã 1 1 : BS si !
{ ‡ i ‡ aX + È Noi T † i T o ĩ 7 T t oO xt 9 nN = = 2 œ ° xt Q ° OQ oO oO oO ® — oO So OQ oO oO +<1qq q91q 1<. 491q
Figure 8.9: Probability of DDR>r as a function T,,,/T for r values from 0.8 to 1.5
Tsys/T
151
Figure 8.10: Probability of TDR>r as a function of T,,;/T for r values from 1.0 to 1.5
While the curves of Figures 8.9 and 8.10 are very informative, it is more convenient for design purposes to rearrange these curves for the purpose of estimating the effects of SSI on the performance of the piers with given target reliabilities. Figures 8.11 and 8.12 show the probabilities of DDR>r and TDR>r as functions of r for bridge piers with T;,,/T from 1.02 to 1.3. The advantage of such presentation of the results of reliability analyses is that for a specific pier with known T,,,/T, the value of r can be found for the desired level of confidence. For example, the curves of Figure 8.12 can be used to obtain the value of r for a bridge pier with T,,;/T of 1.06 with a corresponding confidence of 80%. As shown in this figure, the value of r associated with 20% probability of TDR>r obtained from the curve for Tsys/T=1.06 is 1.23. This means that if the total displacement of the bridge pier is obtained from the analysis of the corresponding fixed-base pier, then the effects of SSI can be accounted for by multiplying this fixed-base displacement by r=1.23 to obtain the total displacement of the pier with 80% confidence.
0.5 m— 1
" Ị Tsys/T = 1.02
ơ_ ; - = = -Tsys/T = 1.06
0.4 7n AN TTT Ts Tsys/T = 1.10
; Tsys/T = 1.15
Kk 03- ! ơ :Tsys/T = 1.20 |
me h Tsys/T = 1.30
Qa 1 t !
2
8 0.2 T ' 1
oy i I
' |
0.1 4
0.0 1
0.7 0.8 0.9 1.0 L1 1.2 1.3 1.4 1.5 1.6
Figure 8.11: Probability of DDR>r as a function ofr for Tạy¿/T of 1.02 to 1.3
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1.0 7 1 Tsys/T = 1.02 - = = 'Tsys/T = 1.06 a romonmonee TSYS/T = 1,107
Tsys/T = 1.15
ằ ô ô ôTsys/T = 1.20
| Tsys/T = 1.30}
0.8 bo - 23 n4en reer e ten ne- ee
Prob TDR>r
1.8 2.0
Figure 8.12: Probability of TDR>r as a function ofr for T;y/T of 1.02 to 1.30
To better demonstrate the proposed practical application of the above SSI modification curves, a bridge example, shown in Figure 8.13, is considered. This bridge has simply supported spans with soil-foundation-pier systems similar to those of this study. It is assumed that the input motions used in this study describe the seismic hazard at the site of the bridge. To design the supports of the deck, the support length must be estimated which is a parameter sensitive to the relative displacement of the piers with respect to each other. The relative displacement of the piers with respect to each other, on the other hand, is described by the total displacements of the two piers which are demand parameters that get affected by SSI. Therefore the effects of SSI on the total displacements of the two piers of the bridge must be estimated. For a simple bridge like this bridge, however, performing SSI analysis is likely not justifiable due to practical constraints and therefore a simplified method can be of great help. This task can be quickly performed with the availability of curves such as those of Figure 8.12 which can be used to obtain SSI modification factors to modify the demands obtained from the analysis of the fixed-base piers. The level of confidence in the modification factor can be chosen by the designer. For instance a confidence level of 80% results in r = 1.23 for the
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pIer with Tsy/T = 1.06 and mì r = 1.62 for the pier with T;y/T = 1.2. Having obtained the values of r for the two piers, the total displacement of the piers with SSI can be calculated from the displacements of the fixed-base piers.
Simply Supported Span
Tays/T = 1.06
Figure 8.13: A bridge example with simply supported spans and pile-supported piers on soft soil
The curves of Figures 8.11, called here “performance-based SSI assessment diagrams for ductility demand” and the curves of Figure 8.12, called here “performance-based SSI assessment diagrams for total displacement demand”, have a number of interesting features that make them appealing for performance-based assessment of the effects of SSI on the response of simple bridges:
e One feature is the explicit consideration of the dispersions of the database of demands that was used to generate these curves. Rather than crudely using mean or median values with unknown levels of reliability, these curves provide the flexibility of choosing the level of confidence in the estimated modification factors. Therefore, the designer can either use a uniform level of reliability for all performance objectives or can use different levels of reliability for different performance objectives tailored for a specific project.
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These curves account for the uncertainties of the system parameters, such as the natural period, that are used to describe the system. The variability of the system parameters must be decided upon prior to generating the curves. Various families of curves can be constructed to account for various levels of uncertainty in the system parameters so the designer can have the flexibility of choosing the level of uncertainty in the estimated system parameters.
In the examples presented here, only system periods, with and without SSI, were considered to construct the curves. However, this can be extended to other system parameters, or combination of system parameters, when constructing the curves.
Constructing the curves as functions of Ts,s/T can expand the domain of the applicability of these types of curves for various combinations of piers, foundations and soils that are different from those used to generate the curves, but have similar T;y./T characteristics. This possibility must be investigated by performing nonlinear dynamic analyses of various pile-supported bridge pier systems on soft soils. If similar behaviour of systems with similar Tsy./T is observed, then such curves become even more appealing for implementation in performance-based design codes for typical highway bridges. Simplified reliable estimation of Tsy;, however, remains a challenge which must be addressed if such curves are to be implemented with a design code format.
If the input ground motions used to construct the curves are all selected to represent a specific level of seismic hazard, then families of curves can be constructed for various levels of seismic hazard. Merits of such selection of ground motions as opposed to the selection criteria used here must be investigated. If the effects of SSI on the response of the piers are not strongly dependent on the amplitude of the input ground motions, then separating these curves for various levels of seismic hazard is not necessary. A preliminary observation of results obtained in this study suggests that DDR and TDR are to
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some degree dependent on the amplitude of ground motions. Figures 8.14 to 8.16 show the relationships between DDR and spectral acceleration, spectral velocity and spectral displacement of the surface motions at the fixed-base period of the piers and illustrate that SSI becomes more effective for stronger ground motions.
Further discussion on this topic is beyond the scope of this dissertation and is recommended for future research.
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T=043s
add
SA (0.3) (g)
T=0.6s
add 15
SA(0.6) @)
T=0.8s
aad 15
SA(0.8) (g)
Figure 8.14: Correlation between DDR and site surface spectral acceleration at the natural period of the fixed-base pier
157
1.5
1ùqaq
052-
0.0
20.0 60.0 0.0 40
SV(0.3) (em/s) 3
20.0 10.0
0.0
15
1.0 +
add
0.5 ee ae
0.0 40.0 60.0 80.0 100.0 120.0
SV(0.6) (cm/s)
20.0 0.0
20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 0.0
SV(0.8) (cm/s)
Figure 8.15: Correlation between DDR and site surface spectral velocity at the natural period of the fixed-base pier
158
15.0 20.0 10.0
SD(0.8) (cm)
159
5.0
eS N T _ 1 ! | . t = : = i t ‹ ! EL—--—=—=—~=—-—--~l--- protec . | ° e ' od e : = Se Đ Ệ yo: o Oo ie .. ` — : ' — “ oO a : le ơ ` ` œ °° aa we eR tO S Ss S So — ! =< > see a I a 0 - : n E~ wn e .* , ' te eon? o e : | o + e * ei . ° ' Ln ee a fe e
e! S 1 e + S "để sẻ ` 4% i ld ° eô .. “ sộ : = v Oo w) xé Se “› ° _ Qo Q _ Qo add aq 0,0
Figure 8.16: Correlation between DDR and site surface spectral displacement at the natural period of the fixed-base pier
8.5 Summary
The data obtained from the nonlinear dynamic analyses were processed probabilistically to explicitly account for the dispersion of results and the uncertainties involved in the estimated system parameters, so that accurate conclusions in regard to the effects of SSI on the response of the bridge pier could be drawn. In summary, the following was observed:
e The probability of SSI increasing the ductility demand of the piers increases with increasing period of the piers or with decreasing period ratio (Tsys/T) of the system. The increase in ductility demand is more that about 25% for systems with period elongation of less than 5% (Tsys/T<1.05) or for piers with periods of greater than 1.0 second. This is a considerable probability and emphasized that SSI should not always be ignored as a conservative assumption in the design of the pile-supported bridge piers.
e The probability of SSI increasing the total displacement demand of the piers is higher for piers with lower natural period or with higher system period ratio (Tsys/T). There is at least about 60% chance of larger total displacement demands of the pile-supported bridge piers compared to their corresponding fixed-base piers which confirms the necessity of attention to the design of displacement sensitive bridge components when SSI is involved.
Results of the reliability analyses were then presented in a format useful for performance- based assessment of the effects of SSI on the response of pile-supported bridge piers by using their corresponding fixed-base response. The application of the proposed methodology was demonstrated by a simple example and the appealing features of the proposed methodology for performance-based design of bridges were discussed and suggestions were made for future research to explore the implementation of the proposed methodology.
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9 EVALUATION OF NONLINEAR STATIC PUSHOVER ANALYSIS FOR DEMAND ESTIMATION INVOLVING SOIL-STRUCTURE INTERACTION
Nonlinear static pushover analysis has received much attention in recent years for seismic demand estimations of structures. The idea behind pushover analysis is to determine the inelastic deformations and the inelastic forces of various components of a system by pushing the structure monotonically at a reference point, e.g. the centre of mass of the bridge superstructure, to the expected total (global) displacement of that reference point under earthquake loading (target displacement). This infers that the target displacement must be determined before performing the pushover analysis. This is normally done by performing linear elastic analysis of the structure to obtain the elastic displacement and then to estimate the inelastic displacement by using equal displacement rule, or other modification factors, or any other approximate method (e.g. FEMA-356 2000, ATC-49 2003).
Assuming that correct target displacements are estimated, pushover analysis provides an exact representation of the response of single degree of freedom (SDOF) systems, such as the bridge piers of this study with fixed base (without SSI). However, for multi-degree of freedom systems, such as multi-story frame structures, or bridge piers with flexible base, there might be shortcomings associated with this method, and the pushover prediction of the inelastic forces and the inelastic deformations of systems components might not match those obtained from nonlinear dynamic analysis. This has been the subject of research for building structures (with many degrees of freedom) but not much for bridge piers with simpler structural form that can be represented by SDOF systems. The simple
161
structural form of bridge piers eliminates the concerns in regard to the shortcomings of pushover analysis for this type of structures. However, adding the flexibility of the base of the piers (due to SSI) makes the flexible-base pier a more complicated system. Hence, the objective of this chapter is to evaluate the accuracy of the pushover analysis by comparing the pushover prediction of demands with those obtained from the nonlinear dynamic analyses. Note that the bridge piers of this study have the simplest form and are SDOF systems when their base is fixed. Therefore, since pushover analysis represents the exact behaviour of the fixed-base piers, any difference between the results obtained from the analyses of the flexible-base piers is attributed to the effects of SSI on the response of the system.
9.1 Base Shear Demands
To perform the pushover analyses, the same numerical models of the pile-supported bridge piers that were used to perform the nonlinear dynamic analyses were used. Very low amplitude constant velocities (1e-6 m/s) were applied horizontally at the centre of mass at the top of the piers, as shown in Figure 9.1, and the resulting base shears and total displacements of the piers were tracked. The force-displacement curves thus obtained from the pushover analyses were plotted along with the demands obtained from the nonlinear dynamic analyses. Resulting plots are shown in Figures 9.2 to 9.7 for the piers with fixed-base periods of 0.3 s to 2.0 s. Note that the displacements in these figures are the total displacements (global displacements) of the piers as shown in Figure 5.3 and they are normalized with respect to the height of the piers.
Figure 9.2 shows that the shear forces predicted by the pushover analysis are different than those obtained from the nonlinear dynamic analyses of the pier with T = 0.3 s. It can be observed in this figure that for target displacements of about 0.4% of the piers height and above, the shear forces predicted by the dynamic analyses are lower than that of the pushover curve. This is because larger deformations are occurring in the soil when the system in analysed dynamically as apposed to when it is analysed statically by pushover analysis. Hence, the pushover analysis predicts larger contribution of the deformation of the pier in the total displacements compared to that predicted by the dynamic analysis,
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which results in higher predicted shear forces for a given target displacement. Same difference can be observed for the piers with T = 0.6 s (Figure 9.3), but with less deviation of the results obtained from pushover and dynamic analyses. As can be observed in Figures 9.4 to 9.7, for piers with periods greater than 0.6 s, plausible predictions are made by the pushover analyses. Thus, it is concluded that as SSI becomes more effective with decreasing natural period of the pier, the predictions of pushover becomes less accurate as it cannot properly account for the dynamic behaviour of the soil when the interaction is significant. Note that the observations here are consistent with the observations in previous chapters in regard to the displacements of the system and the increasing contribution of foundation translation in the response of the piers with decreasing natural periods.
Applied Velocity = 1e-6 m/s ————=>BR
Plastic Hinge
Figure 9.1: Numerical model for pushover analysis
163
2000.00
1500.00 +-
1000.00 ơ
Pier Base Shear (kN)
500.00 +
@ Nonlinear Dynamic Analysis
ẳ —eePUIshover Analysis
0.00 —— + †—
0.0 0.5 1.0 1.5 2.0
Normalized Total Dis placement (% of Height)
Figure 9.2: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.3 s
T=0.6s
2000.00 T T k
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0.0 0.5 1.0 1.5 2.0 2.5 3.0
Normalized Total Displacement (% of Height)
Figure 9.3: Comparison of the force-displacement relationships (including SSD) obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.6 s
164
T=0.8s
2000.00 1 1 T
i Ù
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a °.
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& 500.00 + 1 + i
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0.0 0.5 1.0 1.5 2.0 2.5 3.0
Normalized Total Displacement (% of Height)
Figure 9.4: Comparison of the force-displacement relationships (including SSI) obtained from pushover and nonlinear dynamic analyses of the piers with T = 0.8 s
T=1.0s 2000.00
® Nonlinear Dynamic Analysis
some Pushover Analysis !
— 1500/00 + ; o | +
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0.0 0.5 1.0 1.5 2.0 2.5 3.0
Normalized Total Displacement (% of Height)
Figure 9.5: Comparison of the force-displacement relationships (Including SS]) obtained from pushover and nonlinear dynamic analyses of the piers with T= 1.0s
165
T=1.5s
2000.00 T i t 1 :
e Nonlinear Dynamc Analysis
—=ese DU shover Analysis '
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0.00 1
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Normalized Total Displacement (% of Height)
Figure 9.6: Comparison of the force-displacement relationships (including SSJ) obtained from pushover and nonlinear dynamic analyses of the piers with T= 1.5 s
T=2.0s
2000.00 1 1
@ Nonlinear Dynamic Analysis} -: ; owumme Pushover Analysis
= 1500.00 + +
⁄ — ! Ị : 1 | t . 1 | |
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% † | | I
œ I i i '
œa | i 1 Ị
5
l 1 |
0.00 + T T T T
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Normalized Total Displacement (% of Height)
Figure 9.7: Comparison of force-displacement relationships with SSI obtained from pushover and nonlinear dynamic analyses of the pier with T = 2.0 s
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9.2 Global Ductility Demands
Global ductility demand is another demand parameter used to measure the inelastic response of bridge piers. Global ductility demand is similar to local ductility demand except that instead of using the displacements measured with respect to the base of the pier, the total displacements (global displacements) are used to calculate global ductility demand, as shown in Equation 9.1 (refer to Figure 5.3 for the parameters)
A tot — A. + (A; +Ap)
_ơ (9.1)
AVS Ay+(Ar+Ag)
Bh global =
Estimation of the global ductility demand of a system requires estimation of the total displacement demands (global displacement) A, of the system as well as the system yield displacement (global yield displacement) A¥* which could be estimated by performing pushover analysis. Note that this is another application of pushover analysis which is the estimation of the capacity of structural systems. For capacity estimation purposes, pushover analysis is performed by pushing the structure to large nonlinear deformations, and by plotting the resulting force-displacement curve. This curve is then idealized into a bilinear or multi-linear curve from which the global yield displacement of the system is extracted.
Assuming that total displacement demands are accurately predicted, the objective here is to examine the accuracy of pushover analysis in predicting the global ductility demands.
For this purpose, the total displacements obtained from nonlinear dynamic analyses are used with the system yield displacements obtained from pushover analysis. Resulting global ductility demands are then used to calculate the ratio of the global ductility demands to their corresponding local ductility demands. The mean values of these global- to-local-ductility-ratios are plotted in Figure 9.8. This figure shows that the predicted global ductility demands were higher than their corresponding local ductility demands, especially for the piers with periods of less than 1.0 s. This is an important observation because global ductility demands are expected to be less than local ductility demands (for
167