Optimum Design of Curved Surface Sliders for 2 Minimum Structural Acceleration and Its Sensitivity 3

The design of curved surface sliders (CSS) based on the elastic response spectrum is done 12 by iteration to find the combination of friction coefficient and displacement capacity which satisfies 13 the condition that the maximum horizontal CSS force is equal to the horizontal force of the 14 structure. Although this CSS design is valid it does not necessarily minimize structural 15 acceleration. This paper therefore describes the optimum CSS design for minimum structural 16 acceleration. All valid CSS designs and the optimum CSS design are represented by their associated 17 trajectory in the elastic response spectrum plane which visualizes the optimization problem. The 18 results demonstrate that the optimum CSS design is not obtained at maximum tolerated effective 19 damping ratio. The subsequent sensitivity analysis describes how much the structural acceleration 20 increases if the actual friction coefficient of the real CSS deviates from its optimum design value. 21 The analysis points out that the increase in structural acceleration is approximately one order of 22 magnitude smaller than the deviation in friction. The sensitivity data may be used by structural 23 engineer to determine tolerable deviations in friction coefficient which still results in acceptable 24 structural accelerations. 25


Introduction
Curved surface sliders (CSS) shift the natural period of the primary structure away from the time period range of high seismic energy and augment structural damping by friction damping [1].
For a selected isolation time period the CSS may be designed by the elastic response spectrum method assuming friction coefficient and displacement capacity [2].A valid CSS design is obtained if the maximum longitudinal CSS force is equal to the maximum horizontal force of the primary structure due to the ground acceleration [2].Thus, infinite CSS designs may be obtained by different combinations of friction coefficient and displacement capacity but there is only one combination minimizing structural acceleration.This design freedom led to a variety of investigations on the damping in CSS.Back in 1991 Lai and Song [3] started investigating the impact of CSS damping on structural acceleration, followed by Inaudi and Kelly [4] in 1993.Later the controversy discussion on the role of linearized CSS damping on CSS displacement capacity and structural acceleration was published in 1999 by Kelly [5] and Hall [6].Du and Zhaou [7] used a linearized 2-degree-of-freedom model of the structure with CSS to investigate the optimum damping range of CSS.A first approach towards the optimization of the friction coefficient of CSS was presented by Jangid [8] in 2005 where the optimization criterion was to minimize both top floor acceleration and CSS relative motion for near-fault motion.Depending on the ground motion data the optimum friction coefficients turned out to be between 5% and 15%.Bucher [9] extended this optimization task by also taking into consideration the re-centring condition and Kovaleva et al. [10] further developed this approach to derive optimum CSS parameters for various optimization functions.In the work of Nigdeli et al. [11] a linearized model of the structure with CSS was adopted to derive optimum CSS parameters that minimize structural acceleration with the constraint of a maximum tolerated CSS displacement capacity; a similar approach was used by Kamalzare et al. [12] to keep computational efforts within reasonable limits.
Common to most of the above mentioned studies is that basically optimum results only are shown but not all valid CSS design solutions.Also, the results of these studies are valid for certain ground motion data but not for the entire possible variety of ground accelerations as specified by the elastic response spectra of type 1 and 2 with soil classes A, B, C, D and E [2].This paper tries to fill this gap by first showing the characteristics of all valid CSS designs from which the optimum CSS design for minimum structural acceleration directly follows.In a next step it is shown how the characteristic variables such as friction coefficient, displacement capacity, effective damping ratio, reduction factor, effective time period and re-centring condition of all optimum CSS solutions depend on the selection of the isolation time period.These two first studies are performed for spectra of type 1 and 2 and soil class C. In the third and final section of the paper a sensitivity study is presented for all spectra types and soil classes which describes by how much structural acceleration will deteriorate when the actual friction coefficient of the real CSS differs from its optimum value.This study gives a clear statement on the acceptable tolerance of the friction coefficient of CSS and can be used by structural engineers to determine maximum tolerable deviations in the actual friction coefficient to still guarantee acceptably small structural accelerations.

Optimum Curved Surface Sliders for minimum structural acceleration
Section 2.1 describes the CSS design adopting the linear elastic response spectrum method.The graphical representation of all valid CSS designs in the response spectrum plane is shown and discussed in section 2.2 based on which the optimization routine is presented to obtain the optimum CSS design for minimum structural acceleration response.The optimization results for spectra of type 1 and 2 are given in sections 2.3 and 2.4.

Linear elastic response spectrum method
The linear response spectrum describes the peak acceleration e S normalized by the peak ground acceleration g a as function of the time period T of the structure modelled as single degree-of-freedom (DOF) system whose damping ratio S ζ is 5% (Figure 1) [2].The spectra of different earthquakes and soil classes are defined by their peak ground acceleration g a , their type (1 or 2), their soil parameter S and time periods B T , C T and D T .The acceleration response at time periods Once the spectrum is defined by the soil dynamics experts the CSS design starts by the selection of the targeted isolation time period iso T of the structure with CSS which determines the effective radius eff R of the curved surface of the CSS according to . Then, a combination of friction coefficient μ and displacement capacity bd d of the CSS must be assumed in order to be able to compute the following states (Figure 2): • effective stiffness: • effective time period: • effective damping ratio: • reduction factor: • reduced acceleration response of structure at effective time period: (acceleration response determined from spectrum and multiplied by η )        The location of the optimum CSS design becomes nontrivial when the optimizations are performed for iso T =1.49 s (Figure 6) and iso T =1.50 s (Figure 7).• The optimum CSS design in the typical isolation time period region 3.5 s< iso T <5 s is not obtained from maximum tolerated effective damping ratio eff ζ =30% but from the lower value eff ζ =17.8% evoking η =0.66.

Optimum CSS design for minimum structural acceleration
• The jump in the curves of the shown state variables at iso T =2.

Optimization results for spectrum of type 2 with soil class C
The optimization trajectories associated by their optimum CSS designs for spectrum type if iso T ≥ 1.5 s which explains the jump in the state variables at iso T =1.5 s (Figures 12 and 13).Similar to the results for spectrum of type 1 not the maximum tolerated effective damping ratio eff ζ =30% with associated η =0.55 but eff ζ =17.6% with associated η =0.66 minimizes structural acceleration.
Also, the re-centring condition is fulfilled for all considered iso T and shows the same value as for spectrum of type 1.

Sensitivity of friction coefficient on structural acceleration
This section describes quantitatively by how much the structural acceleration worsens when the actual friction coefficient of the real CSS deviates from its optimum value minimizing structural acceleration.

Sensitivity formulation
The reduced structural acceleration response Please note that this sensitivity analysis does not investigate the impact of deviations in μ on the resulting displacement capacity.It is common understanding that a lower actual friction coefficient of the real CSS than its design value will cause larger relative motions in the CSS as, e.g., visible in Figures 4(b) and 5(b).

Results for spectrum of type 1
The sensitivity curves for spectrum of type     are very similar to those for spectrum of type 1 because the relative deterioration of the structural acceleration depends on the relative change of the friction coefficient.

Summary and conclusions
This paper first presents an optimization routine that derives all valid designs of curved surface sliders (CSS) based on the method of the linear response spectrum.All valid CSS designs are represented by their acceleration trajectory in the elastic response spectrum plane.The fairly flat minimum of the acceleration trajectory reveals that deviations in the actual friction coefficient of the real CSS from its optimum value do not have great deteriorating impact on structural acceleration.
The second part of this study describes how friction coefficient, displacement capacity, effective damping ratio, reduction factor, effective time period and re-centring condition of all optimum CSS solutions for minimum structural acceleration depend on isolation time period.The results for reasonable isolation time periods demonstrate that the optimum CSS, which minimizes structural acceleration, is not obtained at maximum tolerated effective damping ratio of 30% of the CSS but at a significantly lower value.
The third and final part of the paper is concerned with the question by how much the structural acceleration deteriorates when the actual friction coefficient of the real CSS differs from its optimum value.The underlying sensitivity analysis, which is performed for spectra of type 1 and 2 and all soil classes, demonstrates that the relative increase in the structural acceleration is approximately one order of magnitude smaller than the assumed deviation in the actual friction coefficient from its optimum value.The sensitivity results may be used by the structural engineer to define tolerable deviations in the actual friction coefficient from its optimum value such that the resulting structural acceleration response is still acceptably small.
the resulting CSS design -although a valid design -does not necessarily also minimize the acceleration response of the structural.The subsequent section therefore describes a procedure how to obtain all valid CSS designs due to all possible combinations of μ and bd d based on which the CSS design leading to minimum structural acceleration can be identified.

Figure 1 .
Figure 1.Elastic response spectrum including trajectory of all valid CSS designs, limitations due to minimum reduction factor and maximum effective damping ratio and optimum CSS design for minimum structural acceleration.

A 1 .
software program is presented which allows computing any valid CSS design based on different combinations of μ and bd d satisfying S b F F ≈ ; Figure 2 depicts the flow chart of this software program.The software program basically consists of a for-loop where μ is selected based on the assumed friction coefficient range range μ beginning at μ =0.2% and ending at μ =20% with increment of 0.025% and a while-loop that computes bd d for the selected μ such that the relative error of S b F F ≈ is not greater than 0.5‰.For the assumed friction coefficient range with 792 elements this software program computes 792 valid CSS designs.These valid CSS designs are plotted in the spectrum plane in Figure 1 by their reduced normalized structural acceleration time period eff T .Due to the small increment in range μ the plotted points of these valid CSS designs appear as a line, i.e. the trajectory of valid CSS designs.Along this trajectory the design parameters μ and bd d of the valid CSS designs change which is shown by some selected valid CSS designs with associated values of μ and bd d .From this trajectory the following can be observed: a) The trajectory of the valid CSS designs starts at iso eff T T ≈ due to the smallest considered friction coefficient μ =0.2% and then primarily propagates to the "left", i.e. to lower values of eff T due to the increasing values of μ and the acceleration response of the trajectory is reduced due to 0.55 ≤ η ≤ Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 22 January 2018 doi:10.20944/preprints201801.0199.v1Peer-reviewed version available at Geosciences 2018, 8, 83; doi:10.3390/geosciences8030083b) As long as ≤ ζ eff 5% the trajectory of the valid CSS designs is congruent with the non-reduced acceleration response of the spectrum because .1% the trajectory of the valid CSS designs is below the non-reduced acceleration response of the spectrum because 5% ≤ ζ < eff 28.1% results in 0.55 η ≤ <1.d) To the "left" of the vertical dash-dotted line in green due to = ζ eff 28.1% and hence η =0.55 the reduction factor remains at 0.55 despite eff ζ increases up to its maximum tolerated value of 30% due to the increasing μ ; eff ζ =30% is indicated by the blue vertical dash-dotted line.e) There exists one optimum CSS design that is valid ( S b F F ≈ ) and minimizes the structural acceleration response which is highlighted by the red cross on the trajectory.The optimum CSS design is determined by the described software program after the computation of all valid CSS designs from which the pair of μ and bd d is selected that minimizes the structural acceleration response.In the subsequent section 2.3 and 2.4 the optimum CSS designs are presented for spectra of type 1 and 2 and soil class C; the results due to other soil classes are omitted as their influence on the optimization results is little.2.3.Optimization results for spectrum of type 1 with soil class C 2.3.1.Optimum solutions for selected isolation time periods Four optimization trajectories due to four selected isolation time periods are presented.

Figure 3 (
Figure 3(a) depicts the case of a rather unrealistically low isolation time period iso T =1.9 s whereby the entire optimization trajectory lies in the region D eff C T T T < ≤ .The optimum CSS design is obtained at minimum tolerated η =0.55 due to eff ζ =28.1%, i.e. on the green dash-dotted line.For the more realistic isolation time period iso T =3.5 s the main part of the optimization trajectory including the optimum CSS design lies in the region D eff T T ≥ (Figure 3(b)).The optimum CSS design is characterized by eff ζ =17.8% and η =0.662.The optimization trajectory does not show how the state variables μ , bd d , eff ζ , η and eff T change along the trajectory.Therefore, the state variables bd d , eff ζ , η and eff T are plotted as function of the varied μ in Figures 4 and 5 for iso T =1.9 s and iso T =3.5 s.The force displacement loop of the optimum CSS solution is also included to demonstrate that the optimum CSS design fulfils the re-centring condition 25 .0 E / E h s ≥ .

Figure 3 .Figure 4 .
Figure 3. Optimization trajectories for (a) iso T =1.9 s and (b) iso T =3.5 s for spectrum type 1 with soil class C.

Figure 5 .
Figure 5. (a-e) Optimization results for iso T =3.5 s as function of friction coefficient and (f) force displacement curve of optimum CSS design for spectrum type 1 with soil class C.
For iso T =1.49 s the optimum CSS design lies in the region D eff T T < because the optimization trajectory, which shows a first local minimum at D eff T T > in section (ii), drops again in section (iii) generating the global minimum at D eff T T < (Figure 6(b)).The optimum CSS design is characterized by eff ζ =28.1% and η =0.55.In contrast, if iso T =1.50 s is selected the global minimum is obtained at D eff T T > with eff ζ =17.8% and η =0.662 similar to the results obtained for iso T =3.50 s (Figure 7(b)).

PreprintsFigure 6 .
Figure 6.(a) Optimization trajectory for iso T =2.49 s and (b) according close-up for spectrum type 1 with soil class C.

Figure 7 .••
Figure 7. (a) Optimization trajectory for iso T =2.5 s and (b) according close-up for spectrum type 1 with soil class C.
5 s is caused by the fact that the optimum CSS design lies in the region is fulfilled for all optimum CSS design solutions for all considered iso T .

Figure 8 .
Figure 8.(a) Minimum structural acceleration due to optimum selections of (b) friction coefficient and (c) displacement capacity for spectrum type 1 with soil class C.

Figure 9 .
Figure 9. States of optimum solution: (a) effective damping ratio, (b) reduction factor, (c) effective time period, (d, e) maximum horizontal CSS force, (f) re-centring condition for spectrum type 1 with soil class C.
2 and soil class C are depicted in Figures 10 and 11.Since D T =1.2 s for type 2 is much lower than D T =2 s for type 1 the optimum CSS design solution lies in the region eff D T T < if iso T <1.5 s and in eff D T T > Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 22 January 2018 doi:10.20944/preprints201801.0199.v1Peer-reviewed version available at Geosciences 2018, 8, 83; doi:10.3390/geosciences8030083

Figure 10 .
Figure 10.Optimization trajectories for (a) iso T =1 s and (b) iso T =2 s for spectrum type 2 with soil class C.

Figure 11 .
Figure 11.Optimization trajectories for (a) iso T =1.49 s and (b) iso T =1.5 s for spectrum type 2 with soil class C.

Figure 12 .Figure 13 .
Figure 12.(a) Minimum structural acceleration due to optimum selections of (b) friction coefficient and (c) displacement capacity for spectrum type 2 with soil class C.

Figure 14 .
Figure 14.Derivation of sensitivity curves (shown for spectrum type 1 with soil class C and iso T =2.5 s): e S η computed as function of friction coefficient μ as depicted in Figure 14(a) basically shows by how much the structural acceleration response deteriorates (increases) if the actual friction coefficient of the real CSS deviates (being smaller or greater) from its optimum value opt μ .The sensitivity is defined as the relative change in the system's output (relative change in structural acceleration) for given relative change in the system's input (relative change in friction coefficient).Hence, the sensitivity curve is obtained by, first, subtracting the minimum value from the structural acceleration and the optimum value from the friction coefficient, respectively, and, subsequently, normalizing the first term by its minimum value and the second term by its optimum value (Figure 14(b)) structural acceleration may deteriorate by +5% (green dash-dotted line) and then read off directly from the sensitivity curves by how much the actual friction coefficient of the real CSS may differ from its optimum value.For the assumption that opt e e S / S δ =+5% may be acceptable the sensitivity curves with reasonable isolation time periods iso T =3 s to 4.5 s demonstrate that the actual friction coefficient of the real CSS may deviate from its optimum value by at least opt / δμ μ =-39.7% and +55.6%.The tolerances are not symmetric relative to opt μ because of the non-symmetrical sensitivity curves.

Figure 15 .Figure 16 .
Figure 15.Sensitivity of deviation in friction coefficient from its optimum value on relative increase of structural acceleration for (a) iso T =2 s and (b) iso T =2.5 s and spectrum of type 1 with soil class C.

Figure 17 .
Figure 17.Sensitivity of deviation in friction coefficient from its optimum value on relative increase of structural acceleration for (a) iso T =4 s and (b) iso T =4.5 s and spectrum of type 1 with soil class C.

Figure 18 .
Figure 18.Sensitivity of deviation in friction coefficient from its optimum value on relative increase of structural acceleration for (a) iso T =1.2 s and (b) iso T =1.5 s and spectrum of type 2 with soil class C.

Figure 19 .
Figure 19.Sensitivity of deviation in friction coefficient from its optimum value on relative increase of structural acceleration for (a) iso T =1.8 s and (b) iso T =2.1 s and spectrum of type 2 with soil class C.

Figure 20 .
Figure 20.Sensitivity of deviation in friction coefficient from its optimum value on relative increase of structural acceleration for (a) iso T =2.4 s and (b) iso T =2.7 s and spectrum of type 2 with soil class C.

Table 1
together with the results for spectrum of type 1 with soil class C.This table reveals that -for reasonable isolation time periods iso T ≥ 3 s for spectrum of type 1 -the actual friction coefficient of the real CSS may differ from its optimum value by at least

Table 1 .
Sensitivity results for spectrum of type 1 for relative increase of +5% and +2% in structural acceleration.

Table 2 .
Sensitivity results for spectrum of type 2 for relative increase of +5% and +2% in structural acceleration.