Evaluation Supporting Element Stiffness of Fluid Viscous Damper Based on Logarithmic and Energy-Dissipated Spectra
Abstract: This study evaluates the optimal values of supporting element stiffness ( ) attached with linear and nonlinear fluid viscous damper FVDs in elastic systems, represented with Maxwell model consisting of dashpot damper in series with a spring ( ) to achieve the efficiency working of FVDs and reduction in structural responses. The effect of supporting element stiffness remains unclear in prior studies especially; the optimal values of supporting element of nonlinear FVDs are required to achieve the optimal performance of structures. Therefore, the developed numerical algorithm by MATLAB is used for computing high-precision solutions for linear and nonlinear hysteresis FVDs that are typically represented mathematically with a Maxwell model in extensive response history analyses of (SDOF) systems with respect to ground excitations. The emphasis of selection procedures of ( ), based on logarithmic spectra technique represented by pseudo velocity and energy-dissipated spectra, are proposed in this study to obtain the optimal values of ( ). These procedures allow to select of ( ) based on the natural vibration period of structures and velocity power of FVDs to achieve the working of pure FVDs and reduction in the structural responses according FEMA-273. Furthermore, provides simple mathematical model to produce realistic estimates of the peak dynamic response of elastic system equipped with linearization FVDs. Finally, a 20-story structure model equipped with FVD including ( ) was simulated numerically in order to check the suggested values of supporting element stiffness.
Keywords: fluid viscous damper; Maxwell model; supporting element; logarithmic spectra; energy-dissipated spectra.
Introduction
In the past three decades, various types of supplemental damping devices have been utilized in many buildings and bridges for mitigating the hazards posed by wind and earthquake excitation. These devices, when properly designed, enhance the performance of the structure by reducing their dynamic response characteristics as displacement, velocity, acceleration, inter-story drift and base shear force. Don't use plagiarised sources.Get your custom essay just from $11/page
The basic information of using fluid viscous dampers to enhance the seismic performance of buildings have been studied by various previous researchers and manufacturers [Constantinou and Symans 1992; Constantinou and Symans 1993; Reinhorn et al. 1995; Taylor and Constantinou 1995; Ramirez et al. 2002 and Kasai et al. (2003); Black and Makris 2007; Symans et al. 2008; Dong et al. 2015; Taylor 2010]. Viscous damper is working as velocity-dependent damping systems. The advantageous of viscous dampers, the generated forces are out-of-phase with displacement-induced forces within a main structural system under seismic excitation (Constantinou et al. 1998), the temperature dependence of fluid viscous dampers is relatively low and also damper forces is limited when velocity exponent is less than 1 [Kasai et al. 2004b; Symans et al. 2008]. Recent earthquakes have demonstrated the effectiveness of FVDs in modifying the response of structural buildings to controlling structural and non-structural damage [Buchanan et al. 2011; Miranda et al. 2012; Kasai et al. 2013].
The mechanical operation properties of a FVD depend on some of parameters, including the damper velocity power , the damping coefficient and supporting element stiffness both of them are more important to ensure the FVD working. No international current design guidelines and codes give specific methods to identify these parameters.
The velocity power that designates to characterize the damping nonlinearity of FVD has been investigated by several researchers and engineers. This parameter affects the efficiency working of a FVD and its force capacity. A nonlinear FVD with could dissipate 11% more energy per cycle than a linear FVD and 31% more than a nonlinear damper with when subjected to harmonic motion with the same amplitude and the same maximum damping force. Meanwhile, a nonlinear damper with equaling 0.5 had a smaller peak damper force than the other two [Symans and Constantinou 1998]. With current technology, a viscous damper with ranging between 0.15 to 2.0 could be manufactured, but 0.4 to 0.5 is most commonly used for seismic design [Taylor 2010]. Generally, a linear damper is more capable of suppressing higher modes and has least interaction with structural forces that are in-phase with displacements [Duflot and Taylor 2008]. However, a linear damper develops larger damper forces than a nonlinear damper if the velocities are large, a long period structure subjected to intense ground shaking thus it is difficult to control the damper capacity [Lin 2002]. Miyamoto et al. [2010] found that large damper forces in linear dampers under moderate or major earthquakes could exceed the damper capacity. Martinez-Rodrigo and Romero [2003] tried to improve a six-story steel structure using different nonlinear dampers. It was concluded that the peak force of a nonlinear damper would be more than 35% smaller than a linear damper assuming both dampers had an equivalent energy consumption. Geol [2002] investigated the seismic response of one-story, one-way asymmetric linear and nonlinear systems with linear and nonlinear fluid viscous dampers shows that the nonlinear damper leads to only minor (less than 10%) reduction in deformations, base shear, and base torque with long periods more than 0.5 sec. Moreover, the determination of damping coefficient was designed to be proportional to story-shear force and story stiffness [Pekcan et al. 1999; Miyamoto 2010].
The most important parameter is appeared recently that impacts the behavior of the damper is the stiffness of the driving supporting element that is in series with a damper represented by Maxwell model. When the stiffness of the supporting element of FVD is not large enough, its flexibility reduces the deformation of dashpot damper, and generates a damper force that is in- phase with displacement of the structural system. Note: In the stiffness of a large support element with a large fluid chamber, the flexibility of the damper is negligible. Given that the brace contributes most to the flexibility of the dashpot-spring assembly, this study assumed represents the stiffness of the supporting element (brace element or connection device). Extensive research has been conducted by [Kasai et al. 1998; Fu and Kasai, 1998; Kasai and Jodai, 2002; Singh et al. 2003; Kasai et al. 2004; Viola and Guidi 2008; Chen et al. 2011; Londono 2014] to study the effect of supporting element stiffness of FVDs. They mainly focused on finding optimal values of supporting element stiffness of linear viscous damper only.
However, it is unrealistic and probably not necessary to use a very large supporting element stiffness as (brace element and connection device) to insure the damper working effect and cost effective in seismic design. They studied the effect of supporting element stiffness of FVDs on the seismic performance of buildings. Many studies have been conducted to determine an ideal value of that maintains the damper effect while limiting the brace size. However, previous researches have given a wide range of the proposed optimal values of the stiffness of supporting element; it seems not more appropriate to conduct a sensitivity study for any specific building equipped with fluid viscous damper. The conclusion of prior studies are:
- Fu and Kasai [Fu 1996; Fu and Kasai, 1998; Kasai et al. 1998] used the performance curves and identified an optimal brace stiffness to be 10 times that of a frame stiffness in order to achieve the desired damper effect;
- Singh et al. [2003] identified a to be five times of ;
- Chen et al. [2011] used a gradient-based algorithm and demonstrated that should be sized as the first story ;
- Viola and Guidi [2008] suggested a in the order of 45 times that of ; and
- Londono [2014] used a first order filter method to size the minimal brace stiffness based on desired damper efficiency over a predetermined frequency range, and found a to be 5 times the product of damping constant and fundamental frequency ( ).
This paper discusses the hysteresis behavior of linear and nonlinear fluid viscous damper attached in series with supporting element stiffness (brace element or connection device) represented by Maxwell model in elastic system under a set of earthquake excitations. The developed model is applied to quantify the optimal size of supporting element stiffness to achieve optimal performance and cost effective in seismic deign. The proposed numerical solution techniques are implemented in mathematical simulation platform MATLAB and are validated by tests nonlinear viscous damper subjected to harmonic loading and ground motion excitation Kasai et al. [2004b].
Model selection of supporting element is based on logarithmic and energy-dissipated response spectra can provide sufficient information about the dynamic behavior of hysteresis fluid viscous damper FVD represented by Maxwell model. Response history analysis of the SDOF systems having different parameters are performed for a range of periods using pure viscous damper and viscous damper-supporting element assembly as Maxwell model. Comprehensive evaluation, response spectra are performed in terms of the pseudo velocity and energy-dissipation capacity for the selection of optimal supporting element stiffness of FVD. Based on this study, we provide optimal values of supporting element stiffness to achieve the efficiency working of FVD as purely damper and reduction of structural responses according to FEMA-273.
Fluid Viscous Damper – Maxwell Model Procedure
Fluid viscous dampers in moment resisting frames and outrigger systems are typically installed with supporting element as brace element and connection device respectively. These components provide axial stiffness and supporting to the damper even to insure the effect of dampers in structural responses under dynamic excitation. The damper is modeled as a Maxwell element to consider a dashpot pure viscous damper and spring stiffness. Fig.1 gives physical and mathematical models of added components for FVD systems in MDOF and SDOF. The spring accounts for the stiffening property of the damper and its supporting brace element or connection device, and the dashpot for its viscous property.
Methodology of Selection Supporting Element Stiffness
The methodology includes the investigation of seismic responses of elastic structures with linear and nonlinear FVD connected with supporting element represented by Maxwell model subjected to ground motions using numerical analysis of differential equations. A completely novel definition of solving constituent equation of SDOF elastic system equipped with Maxwell model of FVD is proposed based on a set of differential equations of equivalent SDOF system. The numerical analysis is applied to quantify the optimum values of supporting element stiffness connected with linear and nonlinear FVD. Procedures selection are based on the dynamic logarithmic and energy-dissipated spectra can provide generally sufficient information about the dynamic behavior of finding the optimal values of supporting element. Response spectra permit to utilize judgment in determining whether the selection of supporting element meets the intended performance of the damper effect to achieve efficiency working of FVD and cost effective in seismic design.
The first procedure, logarithmic spectra represents relation between the pseudo velocity versus natural period of the system. The target pseudo velocity from the damper-supporting element assembly acts like a Maxwell model corresponded to Eq. (12) is less than or equal the same measures from dashpot pure viscous damper corresponded to Eq. (4) to insure the damper effect. FEMA 273 Recommended Provision in (Chapter 2, Table 2-15) respectively, refer to numerical values of damping coefficients for structures supplemented with damping systems at different damping ratios. The target pseudo velocity from the damper-supporting element stiffness tends to be smaller compared to FEMA 273 Recommended Provision. This investigation of logarithmic spectra is concerned with determination of optimal values of support element stiffness to achieve the efficiency of pure viscous damper, FEMA 273 design approach reduction. Based on this procedure, we provide upper and lower bounds for optimal values of supporting element stiffness. The upper and lower bounds are related to the reduction according to purely viscous damper, FEMA 273 respectively.
The second procedure, energy-dissipated spectra represents relation the energy dissipated by FVD versus natural period of the system. Energy dissipation capacity is a better indicator of performance and more significant in seismic demands. This procedure is based on the comparison of energy-dissipated capacity between pure dashpot viscous damper corresponded to Eq. (7) and damper-supporting element assembly corresponded to Eq. (13).
Verification of the Developed Computer Algorithm Using MATLAB
The developed algorithm is verified with prior studies for the numerical models of nonlinear viscous dampers including axial stiffness under two cases (dynamic sinusoidal loading and ground motion excitation). The validation is conducted with damper component experiments conducted in prior studies [Kasai et al. 2004b] based on a Maxwell model. The nonlinear viscous damper that were tested had damper capacity, = 500, 1000 and 2000 kN, axial stiffness, and a damper velocity exponent, . The nonlinear viscous damper was subjected to dynamic sinusoidal loading with increasing of excitation frequency of 0.02Hz, 0.5Hz and 1.5Hz, respectively. Fig.2 shows the measured hysteretic response of the nonlinear viscous dampers in terms of its damper force-displacement relation for different damper capacities and frequencies. In addition, Fig.3 shows the hysteretic response of the nonlinear viscous dampers with damper capacity = 700 KN subjected to El Centro NS and JMA Kobe ground motion excitations and compares between experimental and simulation by developed algorithm. Figs. (3 and 4) show that the time history analysis model of nonlinear hysteresis viscous damper is well accurate under dynamic sinusoidal or seismic excitation.
Effects of Linear and Nonlinear Pure Fluid Viscous Dampers on SDOF Elastic Structures
A comprehensive parametric study is conducted to evaluate the effects of damping ratio and velocity power of pure FVDs attached in elastic structures. As indicated by Eq. (16), the SDOF responses computed along with the different damping ratio values subjected to 20- earthquake ground motions, 40 time histories. It is seen that the structural relative displacement, pseudo velocity and pseudo acceleration responses are both them reduced if supplemental damping is added for elastic structure Figs (5, 6 and 7). The displacement and velocity responses are always reduced when damping ratio is increased, yet the acceleration response is not more effective with a higher level of damping. Therefore, adding more damping to structures subject to real earthquake motions is not good for acceleration response.
On the other hand, to achieve all seismic demands, the energy dissipation capacity of SDOF structures is also evaluated according to Eq. (7) for different damping ratios subject to earthquake motions as shown in Fig.8. Nevertheless, the energy dissipation capacity is be significantly enhanced if there is more damping involved in.
The effects of FVD nonlinearity on seismic responses of SDOF elastic structures with different velocity power ) are also evaluated. Figs (5, 6, 7 and 8) shown the influence of velocity power of FVD on seismic responses (e.g. relative displacement, pseudo velocity and pseudo acceleration) and energy dissipation capacity. The reduction of seismic responses tend to increase with decreasing values of velocity power . Using nonlinear dampers with a smaller α value are more effective in reducing relative displacement and pseudo velocity, especially for structures predominant with intermediate and long periods as shown in Figs (5 and 6). However, the damper nonlinearity is more effective in reduction of pseudo acceleration at short periods (Fig. 7). Compared the energy-dissipation capacities of FVDs with different velocity power and found that a nonlinear damper with smaller α less than 0.5 have large energy dissipation capacity. Fig. 8 shows the FVDs with velocity power (α = 0.4) is more effective than a linear one, and more effective than a nonlinear damper with equaling to 0.2 and 0.6. As an overall evaluation, an α value in the range of 0.2 and 0.4 is considered to be most effective in controlling structural response, limiting maximum forces and having large energy dissipation capacity in seismic design.
Effects of Linear and Nonlinear Fluid Viscous Dampers as Maxwell Model on SDOF Elastic Structures
In this section, the effectiveness of the elastic systems equipped with linear and nonlinear fluid viscous damper FVD as Maxwell model is assessed through post-processing of the time history analysis results. The assessment is based on selection of supporting element stiffness as (brace element of connection device stiffness) of linear and nonlinear FVD.
The support-element stiffness affects greatly the optimum damper stiffness and the response level due to added FVDs. This stiffness should be considered in the design of sizing of the component of added FVD. We seek to find optimum supporting element stiffness to achieve the efficiency of pure FVDs and lower cost in seismic design. The role of the FVD therefore depended on the structural properties (natural frequency), the ground motion characteristics (excitation frequency), linear and nonlinear damping coefficient and velocity power.
In order to determine the optimal supporting element stiffness of linear and nonlinear FVD needed for a SDOF system, the equivalent-damping ratio is necessary to be evaluated/estimated first. However, due to the existence of linear and nonlinear FVD as Maxwell model, a time history analysis has to be done to obtain the maximum structural response of SDOF that is needed for estimation the optimal supporting element stiffness. In this study, the numerical solution for solving differential equation with different supporting element stiffness (Eq.17) is utilized to evaluate the seismic responses of elastic structures with linear and nonlinear viscous damping subjected to 20-ground motions, 40 time histories.
A novel definition of linear and nonlinear FVD as Maxwell model is proposed based on numerical analysis of differential equations of equivalent SDOF system analysis in this study, which does not need the structural response information beforehand hence can facilitate the determination of optimal supporting element stiffness and quantify the effects of linear and nonlinear FVD. The evaluation of optimum supporting element stiffness of FVD as Maxwell model in elastic system is based on logarithmic and energy dissipation capacity spectra. Under this framework, the structural responses represented by logarithmic spectra (e.g. pseudo velocity) and energy dissipation capacity spectra can be obtained to show the behavior of elastic system equipped with linear and nonlinear FVD as Maxwell model. Finally, in searching for the optimal values of supporting element stiffness of linear FVD, one has to balance between the reduction in seismic response represented by pseudo velocity and the increase of energy dissipation capacity.
The fluid viscous damper effect is largely influenced by the supporting element stiffness as a number of researchers based on only linear FVD has acknowledged (brace flexibility or connection device). In order to investigate the effects of supporting element stiffness in elastic system with linear FVD, response of the system with linear FVD is figured in terms of pseudo velocity and energy dissipation capacity.
To evaluate the effect of supporting element stiffness of linear and nonlinear FVDs represented by a Maxwell damper model, we present and compare several sets of results obtained with these two models. The first model is based the pure FVD according to (Eq.16), and the second model is based on the Maxwell model according to (Eq.17). The effect of supporting element stiffness would obviously depend on the relaxation time parameter of the FVD that characterizes the stiffening behavior of the damper. To evaluate this effect, we present several a set of mathematical results obtained for the different values of velocity powers in the two models based on logarithmic and energy-dissipated spectra. The supporting element stiffness is a function of relaxation time parameter and linear and nonlinear damping coefficient.
We try to find optimum value of supporting element stiffness to achieve the same reduction in structural response according to pure FVD and FEMA-273. In the linear and nonlinear FVD, the mathematical analysis can be divided into three groups with different values of relaxation time parameters. The first group uses lower nonlinearity FVD and the second group uses higher nonlinearity FVD .
Lower nonlinearity FVDs
The first group includes lower nonlinearity FVD that operate as Maxwell model on principles such as velocity power and relaxation time. Figs. (9a and 10a) show the effect of different values of relaxation time in pseudo velocity and comprised with pure viscous damper and [FEMA-273] at different natural periods with different damping ratios. Similarly, Figs. (9b and 10b) show the effect of different values of relaxation time on energy dissipation capacity and comprised with pure viscous damper. As is shown in the Figs. (9 and 10), increasing the damping ratio from 10 to 30% decreases the pseudo velocity and increases the energy dissipation capacity. From the results of Figs. (9 and 10) it is seen that the presence of a series spring in the Maxwell model tends to reduce the structural response and increase the energy dissipation capacity, and this effect is more pronounced for lower values of the relaxation time parameter .
The idealized spectrum is divided logically into three period ranges: the short-period region is called the acceleration-sensitive region; the intermediate period region is called the velocity-sensitive region; the long-period region is called the displacement-sensitive region (Chopra 2007, Section 6.8). When relaxation time equals to 0.05, gives a good estimation in pseudo velocity at intermediate and long-period regions of structures comparing with pure FVD as shown in Figs. (9a and 10a). In addition, achieving the energy dissipation capacity comparing with pure FVD Figs. (9b and 10b). Nevertheless, for short-period region, the relaxation shall be smaller value to achieve the same reduction in pseudo velocity comparing with pure FVD and the more effective value within short-period region is . That are, the expressions ( ) may be appropriate in the selection of supporting element stiffnesss based on the natural period of the structure to achieve the efficiency working of pure fluid viscous dampers FVDs. If the designer needs the same reduction of structural responses according to [FEMA-273], it is appropriate to select the optimum value of relaxation time for intermediate and long-period regions and for short-period region; see Figs. (9a and 10a).
Higher nonlinearity FVDs
Supplemental energy dissipation in the form of high nonlinearity FVDs is often used to improve the performance of structures for seismic design. The optimal amount of relaxation time parameter is needed to determine the realistic values of supporting element stiffness and achieve the desired performance of elastic structures. Through mathematical analysis investigation using logarithmic and energy-dissipated spectra, the relaxation time parameter of Maxwell model required to achieve the optimal performance of elastic structures is quantified. Mathematical analysis is proposed which decisively quantifies the effects of relaxation time parameter of higher nonlinearity FVDs on structural responses (e.g. pseudo velocity) and energy dissipation capacity. The second group shows the effect of relaxation time parameter of Maxwell model with higher nonlinearity FVDs in pseudo velocity and comprised with pure viscous damper and [FEMA-273], and on the energy dissipation capacity at different natural periods with different damping ratios as shown in Figs. (11 and 12).
From the results in Figs. (11a and 12a) it is noted that the optimum values of relaxation time at intermediate and long period regions are within the range to achieve the same reduction in structural response according to pure FVDs, and also achieving the energy dissipation capacity comparing with pure FVD Figs. (11b and 12b). Since the optimum value of relaxation, time shall be smaller than 0.01 to achieve the same reduction of pure viscous dampers at short-period regions Figs. (11a and 12a). The structural systems with natural periods predominant with short-period regions, the brace stiffness shall be relatively large with less than 0.01 or equal to infinity to keep the same reduction of pure FVDs. If the designer needs the same reduction of structural responses according to [FEMA-273], it is appropriate to select the optimum value of relaxation time for intermediate and long-period regions and for short-period region; see Figs. (11a and 12a).
The graphs showed how much the velocity power α affects the response for selection values of relaxation time parameter in determining the supporting element stiffness to achieve efficiency working of fluid viscous dampers FVDs. It was observed that the smallest value of supporting element stiffness, the higher the damper effectiveness. As a brace is simply another spring in series with the damper, its effect was similar to that of the series spring in the Maxwell model. In summary, the supporting element stiffness of FVDs becomes more dependent on velocity power of FVD, natural frequency, excitation frequency of ground motions and the nonlinearity of fluid viscous damping coefficient selected for analysis. They were chosen relaxation time values to quantify supporting element stiffness of FVDs so that they just meet the efficiency of pure FVD and a remarkable effect on reduction in the structural responses according and FEMA-273 at different natural periods.
In this study, we suggest lower and upper bound for relaxation time parameter selection to quantify the supporting element stiffness to achieve structural response reduction. The lower and upper bound are related to the reduction according to pure FVDs and FEMA-273. The supporting element stiffness that make the dampers more effective is provided in the following table:
Estimation Maximum Displacement Generated in Fluid Viscous Damper FVD
By adding supplemental FVD connected with supporting element in elastic systems, it was expected that a linear FVD develops larger damper forces than a nonlinear FVD and the displacements are also large at long period structure subjected to intense ground shaking thus it is difficult to control the damper capacity. Designers need to estimate the explicitly values of maximum amplitude displacement for FVD at different natural periods of building to calculate seismic damper capacity, while also making easy prediction maximum realistic dynamic capacity of the system. Therefore, maximum amplitude displacement of FVD, was calculated based on Eq. (11) under several selected ground motions. Fig.13 shows maximum amplitude displacement of linear and nonlinear FVD attached in elastic systems at different natural periods, supporting element stiffness and velocity power relative to maximum displacement of the system . The structural response was obtained from Eq. (18). These results of maximum amplitude of FVD as shown in Fig.10 are found to agree with the prior studies. From where, the displacement of FVD is reduced gradually with decreasing velocity power of FVD.
Using the data provided in Fig.10 for maximum damper displacement relative the maximum displacement of the system called FVD displacement ratio, this ratio was determined by using the least square exponential fitting estimation. The FVD displacement ratio is a function of the natural period of the system , supplemental damping ratio provided by FVD and velocity power based on the data provided in Fig.13 can be expressed by
Fig.10 shows the estimated FVD displacement ratio by equations (18 and 19) compared with the values provided by the differential equation 9. The estimated values were close to the corresponding nominal values, with a discrepancy of less than 15%.
Numerical Analysis
The approach described above is applied to show the estimated supporting element stiffness of linear and nonlinear FVD in a 20-story shear building achieving the efficiency of FVD and the maximum reduction in the performance functions expressed in terms of drift ratio, inter-story rift ratio and base shear. To validate the optimum supporting element stiffness proposed above, modelling the main structure equipped with FVD represented by Maxwell model in a major finite element (FE) software platform. In this study, Perform 3D is chosen as the general FE software to model the structure. This example building is chosen from Reference [18]. The system considered was a 20-story building, consisting of structural elements (columns, beams, braced-frames), and installation fluid viscous dampers incorporated into the bracing system to reduce its response and improve its seismic performance, its structural properties as the natural period is 3.785 sec. The inherent damping ratio is 2%, and therefore, 20-story MRF with viscous dampers have a total viscous damping ratio , at the fundamental period of vibration equal to 20% Reference [18].
To evaluate the effect of FVD represented by a Maxwell damper model, we present and compare several sets of results obtained with this model. The fluid viscous dampers are installed in the middle bay and supported in a horizontal orientation by inverted V braces. The braces are connected to the FVD by gusset plates as pin connection to achieve the efficiency working of FVD. The height-wise distribution of the linear damping coefficients provided Reference [18]. Thus, the nonlinear damping coefficients is computed according to Eq. (8). To characterize the effect of nonlinearity of FVD on the case-study building, six values of 1.0, 0.8, 0.6, 0.4, 0.2 and 0.1 were selected for this sensitivity study.
To study the effects of supporting element stiffness (brace stiffness) on the performance of the structure, two different brace stiffnesses for lower nonlinearity FVD with velocity powers (α= 1, 0.8 and 0.6), namely, are used, and also two different brace stiffnesses, namely, are used for higher nonlinearity FVD with velocity powers (α= 0.4, 0.2 and 0.1). Using two values of relaxation time parameter for lower and higher nonlinearity FVDS to achieve the lower and higher bounds for reduction in structural responses according to pure FVDs and FEMA-273.
This effect would obviously depend on the linear and nonlinear FVD coefficient and natural frequency of structure that characterizes the stiffening behaviour of the damper at lower and higher frequencies. The nonlinear damping coefficients generally leads to a larger sum of supporting element stiffness (brace stiffness).
Time history analyses were performed using Perform 3D Platform for the design basis earthquake hazard level events according to recommendations of ASCE 41-13. Each hazard level contains 20 three-component records, with two-horizontal components and one vertical component.
Using nonlinear dampers with a smaller α value were more effective in reducing peak drift ratios, especially at the most deformed zones; see Fig.15. The distributions of peak drifts were similar for different cases along building height for lower nonlinearity FVDs (α= 1, 0.8 and 0.6) as shown in Figs. (15a, 15b and 15c). However, the reduction of peak drifts were large for higher nonlinearity FVDs until α= 0.2. On the other hand, the distributions of story-shears (Fig.16) predicted that nonlinear dampers were more effective than linear dampers in reducing story-shears; the case with largest damping nonlinearity (α equals to 0.2 and 0.1) predicted the smallest story-shears. As an overall evaluation, an α value in the range of 0.2 and 0.4 was considered to be most effective in reducing structural responses in seismic design.
In this sensitivity study, a numerical study was also conducted to examine the effect of the optimum supporting element stiffness suggested in this paper on which a FVD is installed. Two values of supporting element stiffness of FVD were examined based on the approach described above, which were selected as a function of linear and nonlinear FVD coefficient and natural frequency of the building. It can be seen from Figs (15 and 16) that the inter-story drifts and story-shears decrease as the brace stiffness increases. Suggested values in this paper for lower nonlinearity FVDs with (α= 1, 0.8 and 0.6) to achieve the upper and lower bound of the reduction according to pure viscous damper and FEMA-273 respectively. It was seen that the reductions of the peak inter-story drifts ratios and story-shears for lower nonlinearity FVDs with upper bound were achieved the same reduction of FEMA-273; see Figs (15a, 15b and 15c) & Figs (16a, 16b and 16c). An observable decrease was observed in the story drift ratios and story-shears distributions along the height of the building when was decreased by double from 0.1 to 0.05 to achieve the lower bound of reduction according to pure FVDs, see Figs (15a, 15b and 15c) & Figs (16a, 16b and 16c).
The values corresponding to suggested cases of supporting element stiffness for higher nonlinearity FVDs with (α= 0.4, 0.2 and 0.1) to achieve the lower and upper bound of reduction according to pure viscous damper and FEMA-273, respectively are also plotted. In Figs (15d, 15e and 15f), the suggested values of relaxation time are shown the reduction for the story drifts. Figs (16d, 16e and 16f), we show the reduction in the story shear for the base-shear. The proposed value of upper bound of relaxation time gave more reduction than FEMA-273 because the FEMA-273 reduction values are based on linear damping neither nonlinear.
Finally, the values of supporting element stiffness (brace stiffness, ) of FVD suggested earlier in this study, have a good reduction in structural responses to achieve efficiency working of pure FVDs and reduction according to FEMA-273.
Conclusions
This paper discusses the implementation of the developed numerical algorithms by MATLAB for the numerical solution of the constitutive equations that describe hysteresis fluid viscous damper FVD represented by the Maxwell model in SDOF elastic systems under harmonic loadings and ground motion excitations use higher-order numerical methods (Runge-Kutta – 4th order expansion). The developed numerical model by MATALAB has given a good estimation for nonlinear viscous dampers represents well the experimental data in prior studies under two cases (dynamic sinusoidal loading and ground motion excitation).
The supporting element stiffness as (brace element or connection device stiffness) in series with the dashpot FVD represented by Maxwell model is generally important to achieve the efficient working of pure FVDs, but the effect of supporting element stiffness remains unclear in prior studies especially in case of linear FVDs. This study based on logarithmic spectra and energy dissipated spectra to quantify the size of supporting element stiffness of linear and nonlinear FVDs in SDOF elastic structures. The values of supporting element stiffness were selected based on the structural responses computed from the damper-supporting element assembly acts like a Maxwell model corresponded to Eq. (18) is less than or equal the same measures from dashpot pure viscous damper corresponded to Eq. (17) to ensure the damper effect and also achieve the same reduction in structural responses according to FEMA 273.
The results showed that structural responses could be significantly reduced by the careful selection of supporting element stiffness attached to FVD. Design considerations, with an emphasis on finding the optimum size of supporting element stiffness to guide the designers, include:
- For lower nonlinearity FVDs (5 < ≤ 1), the optimum lower and upper values of relaxation time equals and respectively to quantify supporting element stiffness size to achieve structural performance for structures predominant with intermediate and long-period regions, for structures predominant with the short-period region. The lower and upper values are related to purely viscous dampers and FEMA-273 respectively.
- For higher nonlinearity FVDs ( ≤ 0.5), the optimum lower and upper values of relaxation time equals respectively to quantify supporting element stiffness size to achieve structural performance for structures predominant with intermediate and long-period regions, for structures predominant with the short-period region
The analysis was extended to examine 20-story steel MRFs equipped with FVDs represented by Maxwell model to show that the suggested supporting element stiffness can be applied straightforwardly. Numerical analysis for a 20-story building, the suggested values of supporting element are sufficiently general to be suitable to elastic structures in case of linear and nonlinear FVDs.