- Email: [email protected]
The Effects of Viscous Damping Modeling Methods on Seismic Performance of RC Moment Frames Using Different Nonlinear Formulations
Accepted Manuscript The Effects of Viscous Damping Modeling Methods on Seismic Performance of RC Moment Frames Using Different Nonlinear Formulations
Negar Mohammadgholibeyki, Mehdi Banazadeh PII: DOI: Reference:
S2352-0124(18)30073-0 doi:10.1016/j.istruc.2018.07.009 ISTRUC 303
To appear in:
Structures
Received date: Revised date: Accepted date:
6 February 2018 14 July 2018 17 July 2018
Please cite this article as: Negar Mohammadgholibeyki, Mehdi Banazadeh , The Effects of Viscous Damping Modeling Methods on Seismic Performance of RC Moment Frames Using Different Nonlinear Formulations. Istruc (2018), doi:10.1016/j.istruc.2018.07.009
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT The effects of viscous damping modeling methods on seismic performance of RC moment frames using different nonlinear formulations Negar Mohammadgholibeyki1 and Mehdi Banazadeh2 1
Associate Professor, Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran, Corresponding Author; Email: [email protected]
CR
IP
2
T
M.Sc. Graduate of Structural Engineering, Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran; Email: [email protected]
Abstract: Nonlinear response history analysis, contributing to seismic performance assessment, is a
PT
ED
M
AN
US
constructive tool for evaluating buildings' behavior and damages due to the earthquake. Applying an accurate viscous damping model constitutes a crucial part in this regard. The seismic response and performance of two reinforced concrete moment frames considering various damping modeling techniques and nonlinear elements are investigated. Two basic approaches to consider nonlinearity are selected: distributed and concentrated plasticity, using force and displacement-based fiber elements and elements that contain two end springs and elastic parts to which zero and modified initial stiffness proportional damping are allocated respectively. Rayleigh damping with mass and four different stiffness matrices are applied in fiber elements. The results of Incremental Dynamic Analysis and loss estimation, considering performance-based earthquake engineering methodology, indicate that Rayleigh damping with mass and initial stiffness overestimates collapse capacity and underestimates performance parameters, while, the model with mass and tangent stiffness with updated proportionality terms shows the opposite trend. In concentrated plasticity models, the more flexible the springs, the more conservative the evaluation of building performance.
CE
Key Words: Rayleigh damping Β· Initial stiffness proportional damping Β· Tangent stiffness proportional
AC
damping Β· Loss estimation Β· Collapse capacity
1
ACCEPTED MANUSCRIPT 1. Introduction
US
CR
IP
T
Performance-based design concept has been developed as a beneficial tool in order to address and reduce risks of damages, casualties, and economic losses resulting from an earthquake. Moreover, it considers such risks in the form of meaningful quantities for decision makers, such as the probability of collapse, loss of life, building repair cost, expected annual loss, etc., thereby applying them in the new building design approaches. According to FEMA P58, nonlinear response history analysis is used to evaluate building response, which is a crucial part of performance assessment methodology, in terms of demands (peak story drifts, peak floor acceleration, etc.)[1]. In this regard, building a realistic and reliable nonlinear model is necessary to gain an accurate estimation of inelastic structural response. An appropriate damping model implementation is considered as a complex facet of nonlinear modeling due to its unknown sources. Several studies have been conducted to evaluate ramifications of using various damping models. However, the most accurate one has not been determined yet, due to the lack of experimental data. Typically, for the sake of simplicity, viscous damping, which is proportional to frequency, in the form of Rayleigh proportional damping model has been widely used in spite of its attendant problems mentioned by researchers hitherto. Damping matrix in Rayleigh type damping model is defined by superposition of mass and stiffness proportional terms, which are πΌπ and π½πΎ respectively [2]. πΆ = πΌπ + π½πΎ
AN
(1)
Charney introduced three approaches for modeling Rayleigh damping when the structure is expected to express nonlinear behavior [3]. These approaches are briefly indicated in the table 1.
A
πΌ0
B
πΌ0
C
πΌπ‘
M
Table 1 Approaches for modeling damping in Charneyβs research Mass proportionality term Stiffness proportionality term Stiffness matrix
ED
Approach
Damping matrix
π½0
πΎ0
πΆ = πΌ0 π + π½0 πΎ0
π½0
πΎπ‘
πΆ = πΌ0 π + π½0 πΎπ‘
π½π‘
πΎπ‘
πΆ = πΌπ‘ π + π½π‘ πΎπ‘
AC
CE
PT
βtβ and β0β indexes for πΌ and π½ show that these proportionality terms are computed based on tangent and initial stiffness matrices respectively. Studies done by Charney on an inelastic five-story structural system have shown that implementation of Rayleigh type damping model based on mass and initial stiffness proportional damping (Approach A) can result in overestimation of damping forces which can mislead analysts. Moreover, using mass and tangent stiffness proportional damping with updated proportionality constants (Approach C) creates almost no excessive damping forces [3]. Hall has also provided examples of nonlinear analysis in which the use of mass and initial stiffness proportional damping has brought about spuriously large damping forces [4]. Bernal assessed the generation of spurious damping forces in structural systems due to the presence of massless DOFs or the DOFs with relatively small inertia, considering the larger number of DOFs essential to build up the stiffness than the mass matrix. Based on Bernal's study, the appropriate remedy to avoid such excessive damping forces, which are indicated to have effects on maximum internal forces rather than displacements, is to assemble the damping matrix by a modified stiffness matrix with the DOFs required for mass matrix and expand it to the full coordinates equal to zero [5]. One of the few studies about damping modeling which includes both experimental and numerical tests is conducted by Petrini et al (2008). In this research, a cantilever reinforced concrete bridge pier was tested via a shaking table. In addition, the numerical model of the pier based on both concentrated (nonlinear hinges at two ends of the pier) and distributed plasticity (with fiber sections) was generated. Damping was modeled by 2
ACCEPTED MANUSCRIPT
M
AN
US
CR
IP
T
Rayleigh formulation with initial and tangent stiffness matrix. Comparison of two series of results derived from the test and numerical models showed that by concentrated plasticity simulation, modeling Rayleigh damping with initial stiffness results in underestimation of displacement demand. On the other hand, using tangent stiffness provides more appropriate results in terms of damping force and displacement. However, in the fiber simulation, the closest results to experimental tests were achieved when no damping was added to the model [6]. Erduran investigated on 3 and 9 story steel frames to evaluate inelastic response of structures modeled by different damping models [7]. As well as previous researchers, Erduran proved that Rayleigh type damping implementation based on mass and initial stiffness matrix can lead the structure to unrealistic damping forces [7]. Additionally, floor accelerations and story drift ratios were derived from nonlinear response history analyses for three different seismic hazard levels using original and reduced frequencies for modeling Rayleigh damping. It was concluded that using diminished frequencies accompanied by mass and tangent stiffness matrix in Rayleigh damping can result in cogent responses due to the decline in damping forces, more realistic estimation of demand parameters, and prevention of suppression of higher modesβ effects [7]. Zareian and Medina, by inelastic time history analysis of one degree of freedom idealized frames, demonstrated that Rayleigh type damping implementation with initial stiffness proportional damping causes responses in which peak displacement demand is underestimated; conversely, peak strength demand is overestimated. In order to address such problem, Zareian and Medina proposed a new approach in which each element is modeled consisting of one elastic element in the middle with a modified initial stiffness proportional allocated damping and two rotational springs, having potential of nonlinear behavior, at two ends to which no damping is assigned. The modified proportionality term is defined below [8]: π½ β² = [(π + 1)/π]π½ (2)
AC
CE
PT
ED
Where π is the ratio of rotational stiffness, related to the inelastic springs, to the elastic part of the modified element. According to Zareian and Medina, when the rigid springs are used (i.e.π½ β² = π½) this formulation brings about numerical problems; therefore, it is not recommended. Moreover, incremental dynamic analysis on a 4-story reinforced concrete moment frame modeled with both the proposed approach and the conventional one based on initial stiffness proportional damping, and comparison of fragility curves were performed. It was confirmed that the proposed approach tends to not only eliminate the aforementioned problems but also avoid underestimation of probability of collapse which is possible when structure is modeled by initial stiffness proportional damping [8]. Jehel et.al performed a comprehensive study on the Rayleigh damping based on the initial and updated tangent stiffness matrices in order to develop a set of beneficial analytical formulas for controlling the damping ratios through the inelastic time history analysis which is highly significant in avoiding spurious damping forces. They indicated that using tangent stiffness may be a straightforward approach for controlling modal damping ratios in spite of difficulties in convergence of the model. However, even if the damping ratios are controlled well, the Rayleigh damping based on both types of stiffness matrices, lacks physical evidence [9]. Apart from the aforementioned studies, Chopra and McKenna have also admitted the same issue about using initial stiffness matrix in Rayleigh damping model through assessment of spurious unbalanced moments, generated in a specific joint of a 20-story steel moment frame, which are so high that are even more than the yield moment of the element [10]. However, through the same analysis, using tangent stiffness matrix was confirmed to be more rational because the unbalanced moment in the same joint tended to be much less than the previous state. Additionally, in [10], sensitivity of inelastic response of the concentrated and distributed plasticity models to the damping modeling methods are investigated. According to this study, concentrated plasticity models are more sensitive to the method by which damping is modeled; on the other hand, 3
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN
US
CR
IP
T
distributed plasticity models provide approximately the same results which means that difference in damping models have roughly no impact on the response of these nonlinear models [10]. Hardyniec and Charney also assessed the effects of considering geometrical nonlinearity in terms of P-delta effects combined with different Rayleigh damping models on the collapse behavior of a four-story buckling resisting moment frame and four-story and eight-story steel moment frames. Damping models based on initial stiffness, tangent stiffness, mass, initial stiffness and mass, and tangent stiffness and mass matrices were considered and fragility curves describing the collapse behavior of such structures were plotted. It is indicated that the damping model based on the initial stiffness provide spurious damping forces and should not be used in nonlinear time history analyses [11]. Moreover, Puthanpurayil et.al proposed a new approach for modeling inherent damping in which the damping matrix is formulated at the elemental level and similar to mass and stiffness matrices, it is assembled in order to achieve a global damping matrix. Both Newmark incremental approach and a revised Newmark total equilibrium approach are used as the implementation scheme for the proposed damping model. In order to evaluate the proposed damping model, performance assessment in terms of Incremental Dynamic Analysis (IDA) is performed on a four story RC frame considering the proposed and conventional damping models and the results are compared [12]. Luco and Lanzi introduced a new way to determine the coefficients of Caughey series representing of a classical damping matrix with the aid of least square approach considering fitting between the polynomial damping ratios and known frequency dependence of them [13]. In another research, they proposed a new inherent damping modeling strategy based on the assumption which considers damping forces proportional to the first derivative of the restoring forces with respect to the time. They verified their model with Rayleigh damping model based on the tangent stiffness [14]. The development of computational systems for modeling nonlinear behavior of structures have provided opportunities to examine various methods of modeling nonlinearity in structures. OPENSEES, developed at pacific earthquake engineering center, is a helpful open-source software framework to that end [15]. In this study, by using OPENSEES, behavior of both distributed and concentrated plasticity models are evaluated. RC moment frame members are modeled by force-based elements, which are formulated by flexibility approach, displacement-based elements defined by stiffness method, beam with hinge elements with determined plastic hingesβ length at two ends of elements, and concentrated plasticity models in accordance with Medina and Zareianβs investigation [8]. In addition, Rayleigh damping with all types of proportionality comprising initial, current, last committed stiffness (the last converged stiffness matrix in each step) and mass proportional damping accompanied by constant -based on initial stiffness- and updated proportionality terms are investigated in distributed plasticity nonlinear models. Until now, studies in terms of damping have emphasized on the evaluation of nonlinear response history analysis results directly; while, in this paper, not only is the focus on inelastic responses, but also seismic performance assessment of RC frames, which can be considered as the outcome of response history analysis, has been highlighted. Inelastic seismic responses of RC frames are extracted from incremental dynamic analysis (IDA), using 22 pairs of ground motion records mentioned in FEMA P695 [16]. Two RC frames are nonlinearly modeled in which, with the aid of OPENSEES facilities, combination of assorted beam-column elements (including force-based, displacement-based, and concentrated plasticity elements) with mass and initial, current, and last committed stiffness proportional damping are used. Then, responses are applied to the performance-based assessment methodology and their effects on the seismic performance parameters such as mean and expected annual loss are appraised.
4
ACCEPTED MANUSCRIPT 2. Design and Modeling Assumptions
IP
T
All results are derived from two reinforced concrete moment frames designed and modeled in OPENSEES program. Modal damping ratio is assumed to be 0.05 for the four-story and 0.035 for eight-story frame. The research conducted by Mander et al. is used in this paper in order to consider the effects of concrete confinement in all models [17]. Some design assumptions and the designed sections are represented in table 2, 3, and 4, respectively. Figure 1 depicts geometrical assumptions by the plan and elevation views of both frames In order to focus on the effects of only damping modeling methods on the numerical seismic assessment, the effects of geometric second-order nonlinearity are not included in structural models.
Table 2 Frame Design Assumptions region Building type Los Angeles (US) Special reinforced concrete moment resisting frame
CR
occupancy commercial
M
AN
US
Table 3 8-Story Frame Sections(mm) beam section story column section (longitude*latitude) number - reinforcement reinforcement 8 7 6 5 4 3 2 1
300*300 - 12Ο18 300*300 - 12Ο18 400*400 - 12Ο18 400*400 - 12Ο18 450*450 - 12Ο18 450*450 - 12Ο18 500*500 - 16Ο18 500*500 - 16Ο18
300*250 - 6Ο18 300*250 - 6Ο18 350*300 - 8Ο18 350*300 - 8Ο18 450*400 - 8Ο18 450*400 - 8Ο18 500*450 - 8Ο18 500*450 - 8Ο18
ED
Table 4 4-Story Frame Sections(mm) beam section story column section (longitude*latitude) number - reinforcement reinforcement
PT
4 3 2 1
300*300 - 12Ο18 350*350 - 12Ο18 400*400 - 12Ο18 450*450 - 12Ο18
350*300 - 8Ο18 350*300 - 8Ο18 400*350 - 8Ο18 450*400 - 8Ο18
CE
Fig. 1 (a) Plan view of 4 and 8-story frames (b) Elevation view of 4-story frame (c) Elevation view of 8-story
AC
By available commands in OPENSEES, elements are modeled considering two major approaches: distributed and concentrated plasticity. Table 5 contains a concise description for each element type used in this research. It was not possible to consider "beam with hinge" elements with distributed plasticity and stiffness approach in this paper since there is no such an option in the OPENSEES program. In order to consider viscous damping in the models with distributed plasticity, Rayleigh type damping with mass and stiffness proportional terms were assigned to each model according to table 6. Moreover, in order to evaluate the effect of the ratio introduced in section 1 by equation (2), denoted as βnβ (the ratio of the rotational end springs' stiffness to the elastic part of element), on nonlinear response and seismic performance of concentrated plasticity modeled frames, three different n ratios are considered (n=1,5, and 10). No rotational masses are present at rotational DOFs of the models with concentrated plasticity formulation; as a result, the appearance of spurious damping moments which is the consequence of mass proportional term in classical Rayleigh damping formulation, is avoided. Figure 2 illustrates all nonlinear and damping models used in this research explicitly.
5
ACCEPTED MANUSCRIPT
Models Concentrated Plasticity Based On Zareian and Medina's Approach
Distributed plasticity Displacement Beam Column
Beam With Hinge
n=1
n=5
n=10
T
Force Beam Column
CR
IP
πΌ0 π + π½0 πΎππππ‘πππ πΌ0 π + π½0 πΎππππππ‘ πΌ0 π + π½0 πΎππ’πππππ‘ πΌπ‘ π + π½π‘ πΎππ’πππππ‘
US
Fig. 2 Concise illustration of damping and nonlinear models used in this paper
Table 5 Description of nonlinear element types used in this research
Force Beam-Column
Beam With Hinge
Zareian and Medina's Proposed Approach
Displacement BeamColumn
Elastic + Zero length (spring) elements
AC
CE
Concentrated Plasticity
PT
Stiffness Approach
Table 6 Damping formulations associated with distributed plasticity models* πΆ = πΌ0 π + π½0 πΎππππ‘πππ πΆ = πΌ0 π + π½0 πΎπππ π‘ ππππππ‘ππ πΆ = πΌ0 π + π½0 πΎππ’πππππ‘ πΆ = πΌπ‘ π + π½π‘ πΎππ’πππππ‘
definition
AN
Flexibility Approach (Using force interpolation functions [18])
type of element
ED
Distributed Plasticity
Origin of elements' definition
M
Type of inelastic behavior in elements
The total elements' length has the potential of inelastic behavior by definition of integration points and fiber sections. A specified length at two ends of the elements, to which fiber sections are assigned, can have nonlinear behavior; other parts are elastic. The whole length of element can have inelastic behavior, however, elements should be divided into parts due to the stiffness method's assumptions, thereby achieving more reliable results. Inelastic behavior is possible to occur in two springs at two ends of elements to which no damping is assigned The middle part of elements to which modified initial stiffness proportional damping is assigned, has elastic behavior [8].
* Descriptions: πΌ0 ,π½0 : Constant proportionality terms based on the original system. πΎππππ‘πππ : Initial stiffness matrix of the structure πΎπππ π‘ ππππππ‘ππ : The last converged stiffness matrix from the previous step of the analysis defined in OPENSEES program πΎππ’πππππ‘ : Tangent stiffness matrix of the structure πΌπ‘ ,π½π‘ : Updated proportionality terms based on tangent stiffness and current frequency of the structure
6
ACCEPTED MANUSCRIPT
CR
IP
T
The main idea of considering three different values for "n" ratio comes from the contention in the literature on choosing the best value for "n". The best theoretical combination would be a perfectly plastic spring with a very high stiffness connected by an elastic beam representing the whole elastic stiffness. Similarly, in the report written by Medina & Krawinkler [19], n=10 is suggested for moment resisting frames which may be due to the point that using large values for "n" is rational which leads to avoiding any rotation in springs when they are not active. In other words, as long as avoiding ill conditioning, "n" should be large provided that the objective is to ensure that the rotational spring in elastic mode acts as a penalty to ensure continuity in rotation across the joint. On the other hand, in the report written by Zareian and Krawinkler [20], the convergence problems and ill conditioning resulting from considering large values for "n" is mentioned as a disadvantage of this modeling method and it is proposed to use n=1 for generic moment resisting frames. In order to evaluate the effect of considering different values for "n" in concentrated plasticity approach on the fundamental period of RC frames, table 7 including the fundamental period of three frames associated with the models with n=1, n=5, and n=10 for both four and eight story frames is drawn.
AN
US
Table 7 Fundamental periods of concentrated plasticity models Frame n=1 n=5 n=10 Four Story 0.711 0.577 0.542 Eight Story 1.017 0.827 0.777
AC
CE
PT
ED
M
The rationale of conducting a comparative study on such frames with a slight difference in their fundamental period comes from a deep look into the plastic hinge models. In concrete and steel structures, the only reliable studies in this regard are those conducted by Haselton et.al [21] and Lignos and Krawinkler [22] in which they mainly offered a set of regression-based backbone curves coupled with hysteresis rules to explicitly incorporate the cyclic deterioration, all extracted from experimental data. So there is a single nonlinear curve for each element consisting of four linear parts from: 1- origin to the yield point, 2-then to capping point, 3- then to residual strength point, and 4- then to ultimate point. The first part of this multi-linear back-bone curve presents effective elastic stiffness; hence, even using a large "n" will not necessarily result in a fundamental period equal to an elastic model. On the other hand, the whole philosophy of inventing "n" is to formulate an elastic element in a series with plastic hinge which enables us to circumvent the problem of ill condition while its stiffness modifier theoretically keeps the initial elastic stiffness matrix intact. While using different n values, actually it is essential to modify the nonlinear back bone curve to which does not perfectly represent initial stiffness of the structure itself. From another aspect, considering different values for "n" will definitely cause the model to face change in fundamental period because the stiffness of the end springs and thereby the whole model will decrease by considering lower values for "n" and as a result, the fundamental period of the frame will have a greater value (π = 2πβπ/π, T is the fundamental period, m and k are mass and stiffness matrices of the structure, respectively).
7
ACCEPTED MANUSCRIPT 3. Analysis Procedure and Evaluation of Results 3.1 Incremental Dynamic Analysis
T
Owing to the point that performance assessment in the form of loss estimation requires both acceleration and displacement demands, peak floor acceleration (PFA) and peak interstory drift ratio were selected as Engineering Demand Parameters (EDP). First mode spectral acceleration (Sa(T1,5%)) was chosen as Intensity Measure (IM). 22 pairs of ground motion records mentioned in FEMA P695, including longitudinal and transverse records, are used to perform IDA analysis [16]. Table 8 indicates a summary of ground motion records used in this paper.
IP
Table 8 Summary of the far-field record set used in this paper [16] Magnitude
Year
Earthquake Name
Recording Station Name
1
6.7
1994
Northridge
Beverly Hills-Mulhol
2
6.7
1994
Northridge
3
7.1
1999
Duzce, Turkey
4
7.1
1999
Hector Mine
5
6.5
1979
Imperial Vally
Delta
6
6.5
1979
Imperial Vally
El Centro Array #11
7
6.9
1995
Kobe, Japan
Nishi-Akashi
8
6.9
1995
9
7.5
10
CR
ID No.
Canyon Country-WLC Bolu
AN
US
Hector
Shin-Osaka
1999
Kocaeli, Turkey
Duzce
7.5
1999
Kocaeli, Turkey
Arcelik
11
7.3
1992
12
7.3
1992
13
6.9
14
6.9
15
7.4
M
Kobe, Japan
Yermo Fire Station
Landers
Coolwater
1989
Loma Prieta
Capitola
1989
Loma Prieta
Gilroy Array #3
ED
Landers
Abbar
1987
Superstition Hills
El Centro Imp. Co.
6.5
1987
Superstition Hills
Poe Road (temp)
7
1992
Cape Mendocino
Rio Dell Overpass
19
7.6
1999
Chi-Chi, Taiwan
CHY101
20
7.6
1999
Chi-Chi, Taiwan
TCU045
21
6.6
1971
San Fernando
LA-Hollywood Stor
22
6.5
1976
Friuli, Italy
Tolmezzo
17
AC
CE
18
1990
PT
Manjil, Iran
6.5
16
In this research, the effects of vertical floor acceleration are neglected; the "Newmark-Beta" approach is used as the time integrator; and the "Newton" method is considered as the solution algorithm for the timehistory analyses. Figure 3 illustrates median IDA curves of both RC frames. In general, a similar trend is observable among curves of various damping models for both four and eight story distributed plasticity modeled frames. It is clear that in the elastic range of response, all curves associated with each distributed plasticity model correspond to each other. However, when the behavior of the structure begins to become nonlinear, as it was previously proven, Rayleigh damping model with initial stiffness proportional term underestimates both drift ratio and peak floor acceleration. Moreover, curves related to damping models with Kcurrent (tangent stiffness)
8
ACCEPTED MANUSCRIPT
IP
T
are approximately consistent with the curves derived from models with Klastcommited (last converged stiffness matrix gotten from previous step of analysis), considering the point that computational efforts for the former model was much more considerable than the latter one. Therefore, it can be concluded that it is thoroughly safe to use Klastcommited in lieu of Kcurrent in order to reduce run time. It can also be seen that the curves related to approach C of table 1 (using tangent stiffness matrix with updated proportionality terms) tend to be the lowest curves, which means that this model leads frames to higher amounts of demand parameters in a specific IM. As it is evident in the curves associated with concentrated plasticity models in figure 3, the higher the n ratio, the less the response in a particular IM level. In other words, implementation of a flexible spring (n=1) leads the analyst to get the highest estimation of demand parameters among all models. In general, it is obvious that the damping model selection, in line with the choice of the structureβs nonlinear formulation, can have a significant impact on the estimation of responses.
CR
3.2 Collapse Assessment
AC
CE
PT
ED
M
AN
US
In this paper, collapse limit-state is defined as when the tangent of the IDA curve gets 20% of the starting (elastic) slope and lower, peak interstory drift ratio reaches 10%, or dynamic instability occurs [23]. Figure 4 depicts collapse fragility curves which illustrate the probability of collapse with respect to each specific spectral acceleration. In this figure, it is evident that using Rayleigh damping model based on mass and initial stiffness matrix underestimates probability of collapse compared to all distributed plasticity-modeled frames as well as using rigid rotational springs in concentrated plasticity models. This means that these two damping modeling strategies may provide less conservative estimation of collapse capacity. In order to discuss the impact of damping modeling selection on the collapse capacities of all models more conveniently in detail, median collapse capacities equivalent to the spectral accelerations in which the probability of collapse is equal to 0.5, are extracted and represented in figure 5. According to this figure, as it was above-mentioned, the highest level of median collapse capacity belongs to Rayleigh damping model based on mass and initial stiffness proportional damping. This result is congruent with the current practice which argues that such damping model overestimates damping forces and underestimates demand parameters such as interstory drift ratio and peak floor acceleration. Consequently, the structure will be analyzed and designed less conservatively by using such damping model. In plain words, using this damping model overestimates buildingβs collapse capacity. Figure 5 also indicates that the difference between median collapse capacities of distributed plasticity modeled frames with Rayleigh damping using Kcurrent and Klast committed are negligible; hence, Klast committed can safely be used in lieu of Kcurrent for modeling Rayleigh damping in OPENSEES program. It can also reduce computational efforts and prevent convergence difficulties. Another considerable point in figure 5 is that using higher n ratio, which means using rigid rotational springs in models, has resulted in larger estimation of median collapse capacity. In other words, implementation of flexible springs (n=1) is a highly conservative way of evaluating collapse capacity and thereby modeling the structure. Another observable point in figure 5 is that in general, distributed plasticity models lead to higher collapse capacity estimation of frames than concentrated ones. This may be because of the difficulty in simulating deterioration due to local buckling of steel reinforcements and also nonlinear interaction of shear and flexural behavior, thereby neglecting such points in fiber sections. Whereas, local behaviors are more convenient to be simulated through concentrated plasticity formulation [24].
9
ACCEPTED MANUSCRIPT
DispBeamColumn
ForceBeamColumn
6
DispBeamColumn
5
ForceBeamColumn
5
Sa (g)
6
0 0.1
Drift
BeamWithHinge
0.1
Drift
Concentrated Plasticity
4
0
0
0
2.5
PFA (g)
BeamWithHinge
5
4
PFA (g)
2.5
Concentrated Plasticity
0 0
0 0
0.1
Drift
(a) DispBeamColumn
ForceBeamColumn
5
PFA (g)
0
0 0
2.5
PFA (g)
2.5
(b)
DispBeamColumn
ForceBeamColumn
5
5
0
ED
Sa (g)
M
5
0.1
Drift
AN
0
US
Sa (g)
CR
6
0 0
T
0
IP
0
0
0.1
drift
BeamWithHinge
0
PT
0
drift
0.1
0 0
4
BeamWithHinge
Concentrated Plasticity
drift
0 0.1
drift
0.1
(c)
4
3
0 0
PFA (g)
Concentrated Plasticity
5
CE 0
AC
Sa (g) 0
0
PFA (g)
4
5
0
0 0
PFA (g)
4
0
PFA (g)
4
(d)
Distributed plasticity
Concentrated plasticity n=1 n=5 n=10
πΌ0 π + π½0 πΎππππ‘πππ πΌ0 π + π½0 πΎππππππ‘ πΌ0 π + π½0 πΎππ’πππππ‘ πΌπ‘ π + π½π‘ πΎππ’πππππ‘
Fig. 3 Median IDA curves for 4-story frame (a) EDP=drift (b) EDP= PFA; and 8-story frame (c) EDP=drift (d) EDP=PFA
10
ACCEPTED MANUSCRIPT
DispBeamColumn
ForceBeamColumn
1
1
1
DispBeamColumn
ForceBeamColumn
Probability of collapse
1
0
BeamWithHinge
Concentrated Plasticity
1
0
Sa (g)
0
7
7
BeamWithHinge
1
Sa (g)
0
7
Concentrated Plasticity
1
US
CR
Probability of Collapse
1
Sa (g)
0
7
IP
Sa (g)
0
T
0
0
0
Sa (g)
0
7
0
Sa (g)
0
7
(a) Distributed plasticity
0
Sa (g)
0
AN
0
7
0
Sa (g)
7
(b) Concentrated plasticity
πΌ0 π + π½0 πΎππππ‘πππ πΌ0 π + π½0 πΎππππππ‘ πΌ0 π + π½0 πΎππ’πππππ‘ πΌπ‘ π + π½π‘ πΎππ’πππππ‘
ED
M
n=1 n=5 n=10
PT
Fig. 4 Collapse fragility curves for (a) 4-story (b) 8-story frames
3 2 1 0
4story
CE
Sa (g)
4
8story
AC
DispBeamColumn Ξ±M+Ξ²Kinit
4story
8story
4story
ForceBeamColumn
Ξ±M+Ξ²Kcommit
8story
BeamWithHinge
Ξ±M+Ξ²Kcurr
Ξ±tM+Ξ²tKcurr
4story
8story
Concentrated Plasticity n=1
n=5
n=10
Fig. 5 Comparison of median collapse capacities for all models
From another aspect, the comparison of standard deviation of median collapse capacities regarding all four damping models for each nonlinear element type as a measure of variability in terms of both building height and nonlinear elements themselves is discussed. At first, the sensitivity of nonlinear elements to damping modeling method is assessed when the building height increases (comparison between 4 and 8-story frames). As it is indicated in the table 9, among distributed plasticity models, standard deviation of collapse capacities for displacement- beam-column elements has been increased for 8-story frame compared to 4-story one, while this quantity for force-beam-column element is reduced; also, the difference of such quantity between 4 and 8-story frames for beam-with-hinge element is
11
ACCEPTED MANUSCRIPT
T
negligible. This means that displacement-beam-column elements are more sensitive than other distributed plasticity element types to the damping modeling method when the buildingβs height increases. Secondly, standard deviations for one frame in terms of different nonlinear formulations is analyzed. According to table 9, the highest amount of standard deviation belongs to concentrated plasticity models, tending to have a high sensitivity to the n ratio. As a result, adjusting n ratio in this damping modeling technique can have a considerable impact on collapse capacity estimation of the building. Moreover, among distributed plasticity models, elements formulated based on flexibility method have the lowest standard deviation which demonstrates that these element types are less sensitive to the Rayleigh damping modeling strategy.
CR
IP
Table 9 Standard Deviation of median collapse capacities of four damping models element type 4-story 8-story Displacement Beam Column 0.190 0.236 Distributed Force Beam Column 0.191 0.129 Plasticity Beam With Hinge 0.161 0.171 Concentrated Plasticity 0.317 0.279
AC
CE
PT
ED
M
AN
US
Considering the assessment of collapse mechanism in frames modeled this study, it has been observed that each frame exhibits approximately the same collapse mechanism for all 44 ground motions. As a result, four frames as samples, are selected to be depicted under the transverse component of Manjil ground motion record. Figure 6 visually represents the onset of collapse in 4 and 8 story frames with two selected damping models which are based on distributed plasticity (FBC-Rayleigh damping with mass and initial stiffness proportional damping) and concentrated plasticity formulations (n=1). According to figure 6.b and 6.d, concentrated plasticity models (whose sample is chosen to be the model with n=1) commonly show the mechanism of collapse of second and third floors. Occasionally, in some cases, the plastic hinges begin to form in exterior columns, but typically, the mechanism of collapse of floors were observed for concentrated plasticity modeled frames. The typical collapse mechanism of frames simulated by distributed plasticity formulation is observable in figure 6.a and 6.c. Due to similarity of the collapse mechanism among frames with distributed plasticity formulation, the frame with force-beam-column elements and Rayleigh damping based on mass and initial stiffness proportional damping is selected as a representative sample for all 12 frames of this group for each 4 and 8 story frames.
12
IP
T
ACCEPTED MANUSCRIPT
(b)
CE
PT
ED
M
AN
US
CR
(a)
(c)
(d)
AC
Fig. 6 Incipient collapse shape for a) 4 story frame (FBC-Ξ±M+Ξ²Kinit), b) 4 story frame (n=1), c) 8 story frame (FBCΞ±M+Ξ²Kinit) and d) 8 story frame (n=1)
One of the important parameters in performance assessment in order to quantify building collapse risk is annual probability of collapse. It can be derived by means of mean annual frequency of collapse (Ξ»c) which is influential in combination of seismic hazard curve of the building site and fragility curve which culminates in calculating annual probability of collapse [25]. Figure 7 illustrates two hazard curves used in this research for 4 and 8-story frame drawn by means of available database on USGS website [26].
13
Annual Frequency of Exceedance
ACCEPTED MANUSCRIPT 0.04
0.016
0.03
0.012
0.02
0.008
0.01
0.004
0
0 0
0.5
1
1.5
Sa (g)
2
2.5
3
3.5
0
0.5
(a)
1
1.5
Sa (g)
2
2.5
3
(b)
T
Fig. 7 Hazard curves for (a) 4-story (b) 8-story frames
M
3.20E-04 1.60E-04 8.00E-05 0.00E+00
PACT
SP3
DispBeamColumn
PACT
SP3
ForceBeamcolumn
Ξ±M+Ξ²Kcommit
Ξ±M+Ξ²Kcurr
CE
Ξ±M+Ξ²Kinit
ED
2.40E-04
PT
Annual Probablity of Collapse
AN
US
CR
IP
In this research, performance assessment is conducted by two relevant programs: The Performance Assessment Calculation Tool (PACT) which has been introduced in FEMA P58 [27], and Seismic Performance Prediction Program (SP3) introduced by Haselton and Baker Risk Group [28]. Although PACT has been widely used by researchers and has produced acceptable results, SP3 has some advantages over PACT that make it more user-friendly. For instance, the hazard curve is accessible online and the user has to only enter the buildingβs location into the program; and the building components can be defined automatically and conveniently. For the sake of brevity, annual probabilities of collapse derived from both programs are compared for the 4-story frame in figure 8 as an example. The performance assessment results shown in what follows, are the ones which are derived from SP3 program. As it is evident in the figure 8, there is a negligible difference between annual probabilities of collapse derived from the two programs. This may be due to the little difference between the algorithms used by these programs and also because of the point that hazard curves available on SP3 database are more accurate than the hazard curves used in PACT into which have to be entered manually.
PACT
SP3
BeamWithHinge Ξ±tM+Ξ²tKcurr
PACT
SP3
Concentrated Plasticity n=1
n=5
n=10
Fig. 8 Comparison of results derived from PACT and SP3 programs for the 4-story frame
AC
Annual probability of collapse for 4 and 8-story frames are represented in tables 10 and 11 respectively. In order to compare all models conveniently, column chart is drawn in figure 9. According to this figure, selecting n=1 in concentrated plasticity models leads to a considerable increase in annual probability of collapse, i.e. a conservative analysis of the building compared to other models. Additionally, among distributed plasticity models, Rayleigh damping with mass and tangent stiffness matrix coupled with updated proportionality terms shows higher annual probability of collapse. Similar to previous results, Rayleigh damping with mass and initial stiffness proportional damping provides the lowest annual probability of collapse, thereby the least conservative estimate of this parameter.
14
ACCEPTED MANUSCRIPT Table 10 Annual Probability of Collapse for 4-story frame Distributed plasticity Concentrated plasticity
Model
Ξ±M+Ξ²Kinit
Ξ±M+Ξ²Kcommit
Displacement Beam Column Force Beam Column Beam With Hinge n=1 n=5 n=10
0.000052 0.000045 0.000071
0.000081 0.000064 0.000096
Ξ±M+Ξ²Kcurr
0.000081 0.000066 0.000096 0.000277 0.000132 0.000086
Ξ±tM+Ξ²tKcurr 0.000102 0.000092 0.000126
Table 11 Annual Probability of Collapse for 8-story frame
Ξ±M+Ξ²Kcommit
Displacement Beam Column Force Beam Column Beam With Hinge n=1 n=5 n=10
0.000006 0.000014 0.000015
0.000014 0.000023 0.00003
0.0003
IP
T
0.000015 0.000021 0.000027 0.000128 0.000037 0.000034
Ξ±tM+Ξ²tKcurr 0.000018 0.000024 0.000031
Ξ±M+Ξ²Kinit Ξ±M+Ξ²Kcommit Ξ±M+Ξ²Kcurr Ξ±tM+Ξ²tKcurr concentrated
0.00012
0.0002
AN
0.00008
0.0001
0.00004
0
M
0
(a)
ED
Annual Probability of Collapse
Ξ±M+Ξ²Kcurr
US
Concentrated plasticity
Ξ±M+Ξ²Kinit
CR
Distributed plasticity
Model
(b)
PT
Fig. 9 Annual Probability of Collapse for (a) 4-story (b) 8-story frames
3.3 Mean Loss
AC
CE
In this part, the mean repair cost of buildings corresponding to each intensity level is demonstrated. This is obtained through definition of fragility and consequence functions of all building components and summation of their repair costs. If collapse occurs for a building, the repair cost will be commensurate with its determined replacement cost [1]. Figure 10 illustrates mean repair cost curves for both frames. According to figure 10, among all distributed plasticity models, mass and initial stiffness proportional damping model has brought about the least estimation of normalized mean repair cost in a specific intensity level; while, Rayleigh damping based on mass and tangent stiffness matrices with updated proportionality terms has the highest evaluation of such quantity. Consequently, using the latter damping model leads to more conservative judgment of building repair cost than the former.
15
DispBeamColumn
25
DispBeamColumn
50
20
40
40
15
15
30
30
10
10
20
20
5
5
10
10
0
BeamWithHinge
25
0 0
3.5
3.5
Concentrated Plasticity
25
20
1.75
15
10
10
5
5
1.4
1.75 Sa (g)
3.5
0 2
0
3
1.4 Sa (g)
Sa (g)
US
(a)
10
0
2.8
0
1.2 Sa (g)
2.4
(b)
Concentrated plasticity n=1 n=5 n=10
M
πΌ0 π + π½0 πΎππππ‘πππ πΌ0 π + π½0 πΎππππππ‘ πΌ0 π + π½0 πΎππ’πππππ‘ πΌπ‘ π + π½π‘ πΎππ’πππππ‘
AN
Distributed plasticity
Concentrated Plasticity
20
10 1
2.8
30
20
0
1.4
40
40
0 0
0 50
50
30
0
2.8
BeamWithHinge
60
20
15
0 0
T
1.75
IP
0
ForceBeamColumn
50
20
0 Mean Loss Normalized by Total Replacement Cost (%)
ForceBeamColumn
25
CR
Mean Loss Normalized by Total Replacement Cost (%)
ACCEPTED MANUSCRIPT
Fig. 10 Mean repair cost normalized by total replacement cost for (a) 4-story (b) 8-story frames
CE
PT
ED
Moreover, in the 4-story frame, implementation of various n ratios has led to more varied quantities than the 8-story one. As it is evident in the figure 10.b, curves corresponding to n=5 and n=10 are approximately consistent for the 8-story frame which indicates that using rigid and semi-rigid rotational springs can result in a same evaluation of building repair cost for the 8-story frame. From another aspect, for the 4-story frame, the difference between normalized mean repair costs of n=1 and n=10 at the intensity level equal to 2.65g is roughly 78% because the model with n=1 has collapsed at Sa=2.15g and the repair cost at Sa=2.65g is equal to 100%. Similarly, the same trend is observable at Sa=2.15g for the 8-story frame in figure 10.b. Therefore, using a flexible rotational spring, the analyst will obtain a considerably conservative evaluation of repair cost.
AC
3.4 Expected Annual Loss
The amount of money on average which has to be paid annually to repair building damages likely to occur due to the probable earthquakes is equal to the expected annual loss (EAL). According to Dhakal and Mander [29], this quantity can be measured by integrating the normalized repair cost over the possible hazard frequencies. Table 12 Expected Annual Loss for 4-story frame ($) Distributed plasticity Concentrated plasticity
Model
Ξ±M+Ξ²Kinit
Ξ±M+Ξ²Kcommit
Ξ±M+Ξ²Kcurr
Ξ±tM+Ξ²tKcurr
Displacement Beam Column Force Beam Column Beam With Hinge n=1 n=5 n=10
24341 23563 25326
30630 27513 26853
30560 27671 27076
33417 30986 28743
49189 27417 21686
16
ACCEPTED MANUSCRIPT
Table 13 Expected Annual Loss for 8-story frame ($)
Ξ±M+Ξ²Kcommit
Ξ±M+Ξ²Kcurr
Ξ±tM+Ξ²tKcurr
23186 23721 21363
28548 27304 24273
28842 27431 25315
29987 27900 25673
41827 20848 17589 45000
55000 44000
30000
33000 22000
T
EAL ($)
Concentrated plasticity
Ξ±M+Ξ²Kinit
Displacement Beam Column Force Beam Column Beam With Hinge n=1 n=5 n=10
15000
11000 0
(b)
AN
Fig. 11 Expected Annual Loss for (a) 4-story (b) 8-story frame
US
CR
0
(a)
Ξ±M+Ξ²Kinit Ξ±M+Ξ²Kcommit Ξ±M+Ξ²Kcurr Ξ±tM+Ξ²tKcurr concentrated
IP
Distributed plasticity
Model
ED
M
As it is apparent in tables 12 and 13 as well as figure 11, concentrated plasticity models have the highest and lowest estimation of EAL. It can be concluded that a flexible rotational spring implementation in the model (i.e. n=1) can lead to a highly conservative evaluation of EAL; which is congruent with the assessment of normalized mean losses as well. Moreover, among distributed plasticity models, mass and initial stiffness proportional damping model provides less conservative results compared to other damping models. Table 14, similar to table 9, is drawn to indicate the standard deviations associated with the derived EAL values in order to quantify variation in this parameter.
CE
PT
Table 14 Standard Deviation of expected annual loss of four damping models element type 4-story 8-story Displacement Beam Column 3322 2628 Distributed Force Beam Column 2630 1671 Plasticity Beam With Hinge 1211 1693 Concentrated Plasticity 11848 10740
AC
Considering table 14, it can be seen that the trend observed for this parameter is similar to the trend observed in the median collapse capacity parameter indicated in table 8. However, since the expected annual loss seems to have much more varied values and much sensitive to different damping models, the difference between the standard deviations is much higher. It is evident that concentrated plasticity models have provided more divergent values than distributed plasticity models.
4. Conclusions In this study, the effects of various and currently used damping modeling techniques coupled with implementation of different nonlinear elements on the seismic response and performance parameters are investigated. Distributed plasticity formulation was used, comprising 12 nonlinear elements (displacementbased and force-based) with different Rayleigh damping modeling strategies. Additionally, 3 concentrated plasticity models were developed, including rigid, semi-rigid, and flexible rotational end springs which have no allocated damping and with the modified stiffness proportional damping assigned to the elastic parts of the 17
ACCEPTED MANUSCRIPT elements. These models were applied to 4 and 8-story RC moment frames. Seismic responses of two frames were evaluated by incremental dynamic analysis, then, collapse and performance assessments were carried out in accordance with FEMA P58 by calculating median collapse capacity, annual probability of collapse, mean repair cost, and expected annual loss. The results derived from this research are elaborated in what follows:
T
PT
AC
CE
ο·
ED
M
ο·
AN
ο·
US
CR
ο·
Using Rayleigh damping formulation based on mass and initial stiffness matrices may result in overestimation of collapse capacity and underestimation of performance parameters. This can be considered as a consequence of overestimation of damping forces and underestimation of demand parameters such as story drift by using this damping model which have been presented by researchers hitherto. Among distributed plasticity models, using Rayleigh damping based on mass and tangent stiffness matrices with updated proportionality terms may lead to a higher estimation of building response, lower estimation of collapse capacity, and higher evaluation of performance parameters, leading the analyst to reach a conservative estimate of such quantities. However, it should be noted that due to the necessity for recalculating frequencies and thereby proportionality terms in each step of analysis and utilizing tangent stiffness matrix, this damping modeling method is inefficiently time-consuming. The results associated with Rayleigh damping based on mass and tangent stiffness matrix (Kcurrent) and last committed stiffness (Kcommit) show an approximate consistency. Therefore, considering convergence issues and prolongation of analysis caused by using Kcurrent, it may be judicious to apply Kcommit in lieu of Kcurrent which results in more time-saving method. Concentrated plasticity models represent a high sensitivity to adjusting n ratio (i.e. stiffness of rotational end springs). The comparison between results derived from using various n ratios in concentrated plasticity models demonstrates that the implementation of a flexible spring may lead the analyst to a greatly conservative estimation of performance parameters; therefore, the stakeholder may make an inordinate investment. On the other hand, using a rigid spring brings about higher evaluation of collapse capacity and lower estimation of performance parameters than other concentrated plasticity models. In this paper, two performance assessment programs were used: PACT and SP3. All results obtained from both programs were roughly equivalent. However, SP3 program is more user-friendly because the user is not required to enter nonlinear dynamic analysis results manually (it is possible by uploading an excel file), the building components can be added to the model automatically, etc. Moreover, it seems to be more accurate than PACT because it has the direct access to the USGS hazard database.
IP
ο·
It is significant to notice that this paper is a comparative analytical study considering numerical and analytical approaches for all models of four and eight story RC frames and should be interpreted with caution. Due to the scarcity of experimental data about this topic in the literature, focusing on inherent damping modeling methods and evaluating their effects on the presently widely used nonlinear models have to be performed by numerical and analytical comparative studies. Yet, sophisticated experimental studies on damping modeling methods and their effect on structural systems' response are highly needed. This would enhance the reliability of analytical studies and possibly lead to the determination of the best damping modeling technique.
References [1]
FEMA P58- Federal Emergency Management Agency. Seismic Performance Assessment of Buildings Volume 1- methodology. Appl Technol Counc 2012;1:278. 18
ACCEPTED MANUSCRIPT Rayleigh, Lord. The Theory of Sound, Vol 1. New York: Dover Publications; 1896.
[3]
Charney F a. Unintended Consequences of Modeling Damping in Structures. J Struct Eng 2008;134:581β92. doi:10.1061/(ASCE)0733-9445(2008)134:4(581).
[4]
Hall JF. Problems encountered from the use (or misuse) of Rayleigh damping. Earthq Eng Struct Dyn 2006;35:525β45. doi:10.1002/eqe.541.
[5]
Bernal D. Viscous Damping in Inelastic Structural Response. J Struct Eng 1994;120:1240β54.
[6]
Petrini L, Maggi C, Priestley MJN, Calvi GM. Experimental Verification of Viscous Damping Modeling for Inelastic Time History Analyzes. J Earthq Eng 2008;12:125β45. doi:10.1080/13632460801925822.
[7]
Erduran E. Measuring bias in structural response caused by ground motion scaling. Earthq Eng Struct Dyn 2012;41:1905β19. doi:10.1002/eqe2164.
[8]
Zareian F, Medina RA. A practical method for proper modeling of structural damping in inelastic plane structural systems. Comput Struct 2010;88:45β53. doi:10.1016/j.compstruc.2009.08.001.
[9]
Jehel P, Leger P, Ibrahimbegovic A. Initial versus tangent stiffness-based Rayleigh damping in inelastic time history seismic analyses. Earthq Eng Struct Dyn 2014;43:467β84. doi:10.1002/eqe.2357.
[10]
Chopra AK, Mckenna F. Modeling viscous damping in nonlinear response history analysis of buildings for earthquake excitation. Earthq Eng Struct Dyn 2015;45:193β211. doi:10.1002/eqe.2622.
[11]
Hardyniec A, Charney F. An investigation into the effects of damping and nonlinear geometry models in earthquake engineering analysis. Earthq Eng Struct Dyn 2015;44:2695β715. doi:10.1002/eqe.2604.
[12]
Puthanpurayil AM, Lavan O, Carr AJ, Dhakal RP. Elemental damping formulation: an alternative modelling of inherent damping in nonlinear dynamic analysis. Bull Earthq Eng 2016;14:2405β34. doi:10.1007/s10518016-9904-9.
[13]
Luco JE, Lanzi A. Optimal Caughey series representation of classical damping matrices. Soil Dyn Earthq Eng 2017;92:253β65. doi:10.1016/j.soildyn.2016.10.028.
[14]
Luco JE, Lanzi A. A new inherent damping model for inelastic time-history analyses. Earthq Eng Struct Dyn 2017;46:1919β39. doi:10.1002/eqe.2886.
[15]
OPENSEES. Open System for Earthquake Engineering Simulation. Pacific Earthq Eng Res Center, Univ Calif Berkeley, CA 2002. http://opensees.berkeley.edu/ (accessed March 17, 2016).
[16]
FEMA P695- Federal Emergency Management Agency. Quantification of building seismic performance factors. Appl Technol Counc 2009:421.
[17]
Mander JB, Priestley MJN, R.Park. Theoretical Stress Strain Model for Confined Concrete. J Struct Eng 1988;114:1804β25. doi:10.1061/(ASCE)0733-9445(1988)114:8(1804).
[18]
Neuenhofer A, Filippou F. Evaluation of Nonlinear Frame Finite-Element Models. J Struct Eng 1997;123:958, 966. doi:10.1061/(ASCE)0733-9445(1997)123.
[19]
Medina R, Krawinkler H. Seismic Demands for Nondeteriorating Frame Structures and Their Dependence on Ground Motions. Pacific Earthq Eng Res Center, Univ Calif Berkeley, CA 2004.
[20]
Zareian F, Krawinkler H. Simplified performance-based earthquake engineering. John A Blume Earthq Eng Res Cent 2009:338.
[21]
Haselton CB, Liel AB, Lange ST. Beam-Column Element Model Calibrated for Predicting Flexural Response Leading to Global Collapse of RC Frame Buildings. Report No.03, Pacific Earthq Eng Res Center, Univ Calif Berkeley, CA. 2008.
[22]
Lignos DG, Krawinkler H. Sidesway Collapse of Deteriorating Structural Systems under Seismic Excitations. John A Blume Earthq Eng Res Cent 2009.
[23]
Vamvatsikos D, Cornell CA. The Incremental Dynamic Analysis and Its Application To Performance-Based Earthquake Engineering. Eur Conf Earthq Eng 2002:10. doi:10.1002/eqe.141.
AC
CE
PT
ED
M
AN
US
CR
IP
T
[2]
19
ACCEPTED MANUSCRIPT Deierlein GG, Reinhorn AM, Willford MR. Nonlinear Structural Analysis For Seismic Design: A Guide for Practicing Engineers. NEHRP Seism Des Tech Br No 4 2010:36.
[25]
Eads L, Miranda E, Krawinkler H, Lignos DG. Improved Estimation of Collapse Risk for Structures in Seismic Regions. FIFTHTEENTH WORLD Conf Earthq Eng 2012.
[26]
USGS. United States Geological Survey n.d. https://earthquake.usgs.gov/hazards/ (accessed June 12, 2016).
[27]
FEMA P58- Federal Emergency Management Agency. Seismic Performance Assessment of Buildings Volume 2 - Implementation Guide. Appl Technol Counc 2012;2:357.
[28]
Seismic Performance Prediction Program by Haselton Baker Risk group. SP3 2015. https://www.hbrisk.com/ (accessed October 23, 2016).
[29]
Dhakal RP, Mander JB. Financial Risk Assessment Methodology for Natural Hazards. Bull New Zeal Soc Earthq Eng 2006:91β105.
AC
CE
PT
ED
M
AN
US
CR
IP
T
[24]
20
Negar Mohammadgholibeyki, Mehdi Banazadeh PII: DOI: Reference:
S2352-0124(18)30073-0 doi:10.1016/j.istruc.2018.07.009 ISTRUC 303
To appear in:
Structures
Received date: Revised date: Accepted date:
6 February 2018 14 July 2018 17 July 2018
Please cite this article as: Negar Mohammadgholibeyki, Mehdi Banazadeh , The Effects of Viscous Damping Modeling Methods on Seismic Performance of RC Moment Frames Using Different Nonlinear Formulations. Istruc (2018), doi:10.1016/j.istruc.2018.07.009
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT The effects of viscous damping modeling methods on seismic performance of RC moment frames using different nonlinear formulations Negar Mohammadgholibeyki1 and Mehdi Banazadeh2 1
Associate Professor, Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran, Corresponding Author; Email: [email protected]
CR
IP
2
T
M.Sc. Graduate of Structural Engineering, Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran; Email: [email protected]
Abstract: Nonlinear response history analysis, contributing to seismic performance assessment, is a
PT
ED
M
AN
US
constructive tool for evaluating buildings' behavior and damages due to the earthquake. Applying an accurate viscous damping model constitutes a crucial part in this regard. The seismic response and performance of two reinforced concrete moment frames considering various damping modeling techniques and nonlinear elements are investigated. Two basic approaches to consider nonlinearity are selected: distributed and concentrated plasticity, using force and displacement-based fiber elements and elements that contain two end springs and elastic parts to which zero and modified initial stiffness proportional damping are allocated respectively. Rayleigh damping with mass and four different stiffness matrices are applied in fiber elements. The results of Incremental Dynamic Analysis and loss estimation, considering performance-based earthquake engineering methodology, indicate that Rayleigh damping with mass and initial stiffness overestimates collapse capacity and underestimates performance parameters, while, the model with mass and tangent stiffness with updated proportionality terms shows the opposite trend. In concentrated plasticity models, the more flexible the springs, the more conservative the evaluation of building performance.
CE
Key Words: Rayleigh damping Β· Initial stiffness proportional damping Β· Tangent stiffness proportional
AC
damping Β· Loss estimation Β· Collapse capacity
1
ACCEPTED MANUSCRIPT 1. Introduction
US
CR
IP
T
Performance-based design concept has been developed as a beneficial tool in order to address and reduce risks of damages, casualties, and economic losses resulting from an earthquake. Moreover, it considers such risks in the form of meaningful quantities for decision makers, such as the probability of collapse, loss of life, building repair cost, expected annual loss, etc., thereby applying them in the new building design approaches. According to FEMA P58, nonlinear response history analysis is used to evaluate building response, which is a crucial part of performance assessment methodology, in terms of demands (peak story drifts, peak floor acceleration, etc.)[1]. In this regard, building a realistic and reliable nonlinear model is necessary to gain an accurate estimation of inelastic structural response. An appropriate damping model implementation is considered as a complex facet of nonlinear modeling due to its unknown sources. Several studies have been conducted to evaluate ramifications of using various damping models. However, the most accurate one has not been determined yet, due to the lack of experimental data. Typically, for the sake of simplicity, viscous damping, which is proportional to frequency, in the form of Rayleigh proportional damping model has been widely used in spite of its attendant problems mentioned by researchers hitherto. Damping matrix in Rayleigh type damping model is defined by superposition of mass and stiffness proportional terms, which are πΌπ and π½πΎ respectively [2]. πΆ = πΌπ + π½πΎ
AN
(1)
Charney introduced three approaches for modeling Rayleigh damping when the structure is expected to express nonlinear behavior [3]. These approaches are briefly indicated in the table 1.
A
πΌ0
B
πΌ0
C
πΌπ‘
M
Table 1 Approaches for modeling damping in Charneyβs research Mass proportionality term Stiffness proportionality term Stiffness matrix
ED
Approach
Damping matrix
π½0
πΎ0
πΆ = πΌ0 π + π½0 πΎ0
π½0
πΎπ‘
πΆ = πΌ0 π + π½0 πΎπ‘
π½π‘
πΎπ‘
πΆ = πΌπ‘ π + π½π‘ πΎπ‘
AC
CE
PT
βtβ and β0β indexes for πΌ and π½ show that these proportionality terms are computed based on tangent and initial stiffness matrices respectively. Studies done by Charney on an inelastic five-story structural system have shown that implementation of Rayleigh type damping model based on mass and initial stiffness proportional damping (Approach A) can result in overestimation of damping forces which can mislead analysts. Moreover, using mass and tangent stiffness proportional damping with updated proportionality constants (Approach C) creates almost no excessive damping forces [3]. Hall has also provided examples of nonlinear analysis in which the use of mass and initial stiffness proportional damping has brought about spuriously large damping forces [4]. Bernal assessed the generation of spurious damping forces in structural systems due to the presence of massless DOFs or the DOFs with relatively small inertia, considering the larger number of DOFs essential to build up the stiffness than the mass matrix. Based on Bernal's study, the appropriate remedy to avoid such excessive damping forces, which are indicated to have effects on maximum internal forces rather than displacements, is to assemble the damping matrix by a modified stiffness matrix with the DOFs required for mass matrix and expand it to the full coordinates equal to zero [5]. One of the few studies about damping modeling which includes both experimental and numerical tests is conducted by Petrini et al (2008). In this research, a cantilever reinforced concrete bridge pier was tested via a shaking table. In addition, the numerical model of the pier based on both concentrated (nonlinear hinges at two ends of the pier) and distributed plasticity (with fiber sections) was generated. Damping was modeled by 2
ACCEPTED MANUSCRIPT
M
AN
US
CR
IP
T
Rayleigh formulation with initial and tangent stiffness matrix. Comparison of two series of results derived from the test and numerical models showed that by concentrated plasticity simulation, modeling Rayleigh damping with initial stiffness results in underestimation of displacement demand. On the other hand, using tangent stiffness provides more appropriate results in terms of damping force and displacement. However, in the fiber simulation, the closest results to experimental tests were achieved when no damping was added to the model [6]. Erduran investigated on 3 and 9 story steel frames to evaluate inelastic response of structures modeled by different damping models [7]. As well as previous researchers, Erduran proved that Rayleigh type damping implementation based on mass and initial stiffness matrix can lead the structure to unrealistic damping forces [7]. Additionally, floor accelerations and story drift ratios were derived from nonlinear response history analyses for three different seismic hazard levels using original and reduced frequencies for modeling Rayleigh damping. It was concluded that using diminished frequencies accompanied by mass and tangent stiffness matrix in Rayleigh damping can result in cogent responses due to the decline in damping forces, more realistic estimation of demand parameters, and prevention of suppression of higher modesβ effects [7]. Zareian and Medina, by inelastic time history analysis of one degree of freedom idealized frames, demonstrated that Rayleigh type damping implementation with initial stiffness proportional damping causes responses in which peak displacement demand is underestimated; conversely, peak strength demand is overestimated. In order to address such problem, Zareian and Medina proposed a new approach in which each element is modeled consisting of one elastic element in the middle with a modified initial stiffness proportional allocated damping and two rotational springs, having potential of nonlinear behavior, at two ends to which no damping is assigned. The modified proportionality term is defined below [8]: π½ β² = [(π + 1)/π]π½ (2)
AC
CE
PT
ED
Where π is the ratio of rotational stiffness, related to the inelastic springs, to the elastic part of the modified element. According to Zareian and Medina, when the rigid springs are used (i.e.π½ β² = π½) this formulation brings about numerical problems; therefore, it is not recommended. Moreover, incremental dynamic analysis on a 4-story reinforced concrete moment frame modeled with both the proposed approach and the conventional one based on initial stiffness proportional damping, and comparison of fragility curves were performed. It was confirmed that the proposed approach tends to not only eliminate the aforementioned problems but also avoid underestimation of probability of collapse which is possible when structure is modeled by initial stiffness proportional damping [8]. Jehel et.al performed a comprehensive study on the Rayleigh damping based on the initial and updated tangent stiffness matrices in order to develop a set of beneficial analytical formulas for controlling the damping ratios through the inelastic time history analysis which is highly significant in avoiding spurious damping forces. They indicated that using tangent stiffness may be a straightforward approach for controlling modal damping ratios in spite of difficulties in convergence of the model. However, even if the damping ratios are controlled well, the Rayleigh damping based on both types of stiffness matrices, lacks physical evidence [9]. Apart from the aforementioned studies, Chopra and McKenna have also admitted the same issue about using initial stiffness matrix in Rayleigh damping model through assessment of spurious unbalanced moments, generated in a specific joint of a 20-story steel moment frame, which are so high that are even more than the yield moment of the element [10]. However, through the same analysis, using tangent stiffness matrix was confirmed to be more rational because the unbalanced moment in the same joint tended to be much less than the previous state. Additionally, in [10], sensitivity of inelastic response of the concentrated and distributed plasticity models to the damping modeling methods are investigated. According to this study, concentrated plasticity models are more sensitive to the method by which damping is modeled; on the other hand, 3
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN
US
CR
IP
T
distributed plasticity models provide approximately the same results which means that difference in damping models have roughly no impact on the response of these nonlinear models [10]. Hardyniec and Charney also assessed the effects of considering geometrical nonlinearity in terms of P-delta effects combined with different Rayleigh damping models on the collapse behavior of a four-story buckling resisting moment frame and four-story and eight-story steel moment frames. Damping models based on initial stiffness, tangent stiffness, mass, initial stiffness and mass, and tangent stiffness and mass matrices were considered and fragility curves describing the collapse behavior of such structures were plotted. It is indicated that the damping model based on the initial stiffness provide spurious damping forces and should not be used in nonlinear time history analyses [11]. Moreover, Puthanpurayil et.al proposed a new approach for modeling inherent damping in which the damping matrix is formulated at the elemental level and similar to mass and stiffness matrices, it is assembled in order to achieve a global damping matrix. Both Newmark incremental approach and a revised Newmark total equilibrium approach are used as the implementation scheme for the proposed damping model. In order to evaluate the proposed damping model, performance assessment in terms of Incremental Dynamic Analysis (IDA) is performed on a four story RC frame considering the proposed and conventional damping models and the results are compared [12]. Luco and Lanzi introduced a new way to determine the coefficients of Caughey series representing of a classical damping matrix with the aid of least square approach considering fitting between the polynomial damping ratios and known frequency dependence of them [13]. In another research, they proposed a new inherent damping modeling strategy based on the assumption which considers damping forces proportional to the first derivative of the restoring forces with respect to the time. They verified their model with Rayleigh damping model based on the tangent stiffness [14]. The development of computational systems for modeling nonlinear behavior of structures have provided opportunities to examine various methods of modeling nonlinearity in structures. OPENSEES, developed at pacific earthquake engineering center, is a helpful open-source software framework to that end [15]. In this study, by using OPENSEES, behavior of both distributed and concentrated plasticity models are evaluated. RC moment frame members are modeled by force-based elements, which are formulated by flexibility approach, displacement-based elements defined by stiffness method, beam with hinge elements with determined plastic hingesβ length at two ends of elements, and concentrated plasticity models in accordance with Medina and Zareianβs investigation [8]. In addition, Rayleigh damping with all types of proportionality comprising initial, current, last committed stiffness (the last converged stiffness matrix in each step) and mass proportional damping accompanied by constant -based on initial stiffness- and updated proportionality terms are investigated in distributed plasticity nonlinear models. Until now, studies in terms of damping have emphasized on the evaluation of nonlinear response history analysis results directly; while, in this paper, not only is the focus on inelastic responses, but also seismic performance assessment of RC frames, which can be considered as the outcome of response history analysis, has been highlighted. Inelastic seismic responses of RC frames are extracted from incremental dynamic analysis (IDA), using 22 pairs of ground motion records mentioned in FEMA P695 [16]. Two RC frames are nonlinearly modeled in which, with the aid of OPENSEES facilities, combination of assorted beam-column elements (including force-based, displacement-based, and concentrated plasticity elements) with mass and initial, current, and last committed stiffness proportional damping are used. Then, responses are applied to the performance-based assessment methodology and their effects on the seismic performance parameters such as mean and expected annual loss are appraised.
4
ACCEPTED MANUSCRIPT 2. Design and Modeling Assumptions
IP
T
All results are derived from two reinforced concrete moment frames designed and modeled in OPENSEES program. Modal damping ratio is assumed to be 0.05 for the four-story and 0.035 for eight-story frame. The research conducted by Mander et al. is used in this paper in order to consider the effects of concrete confinement in all models [17]. Some design assumptions and the designed sections are represented in table 2, 3, and 4, respectively. Figure 1 depicts geometrical assumptions by the plan and elevation views of both frames In order to focus on the effects of only damping modeling methods on the numerical seismic assessment, the effects of geometric second-order nonlinearity are not included in structural models.
Table 2 Frame Design Assumptions region Building type Los Angeles (US) Special reinforced concrete moment resisting frame
CR
occupancy commercial
M
AN
US
Table 3 8-Story Frame Sections(mm) beam section story column section (longitude*latitude) number - reinforcement reinforcement 8 7 6 5 4 3 2 1
300*300 - 12Ο18 300*300 - 12Ο18 400*400 - 12Ο18 400*400 - 12Ο18 450*450 - 12Ο18 450*450 - 12Ο18 500*500 - 16Ο18 500*500 - 16Ο18
300*250 - 6Ο18 300*250 - 6Ο18 350*300 - 8Ο18 350*300 - 8Ο18 450*400 - 8Ο18 450*400 - 8Ο18 500*450 - 8Ο18 500*450 - 8Ο18
ED
Table 4 4-Story Frame Sections(mm) beam section story column section (longitude*latitude) number - reinforcement reinforcement
PT
4 3 2 1
300*300 - 12Ο18 350*350 - 12Ο18 400*400 - 12Ο18 450*450 - 12Ο18
350*300 - 8Ο18 350*300 - 8Ο18 400*350 - 8Ο18 450*400 - 8Ο18
CE
Fig. 1 (a) Plan view of 4 and 8-story frames (b) Elevation view of 4-story frame (c) Elevation view of 8-story
AC
By available commands in OPENSEES, elements are modeled considering two major approaches: distributed and concentrated plasticity. Table 5 contains a concise description for each element type used in this research. It was not possible to consider "beam with hinge" elements with distributed plasticity and stiffness approach in this paper since there is no such an option in the OPENSEES program. In order to consider viscous damping in the models with distributed plasticity, Rayleigh type damping with mass and stiffness proportional terms were assigned to each model according to table 6. Moreover, in order to evaluate the effect of the ratio introduced in section 1 by equation (2), denoted as βnβ (the ratio of the rotational end springs' stiffness to the elastic part of element), on nonlinear response and seismic performance of concentrated plasticity modeled frames, three different n ratios are considered (n=1,5, and 10). No rotational masses are present at rotational DOFs of the models with concentrated plasticity formulation; as a result, the appearance of spurious damping moments which is the consequence of mass proportional term in classical Rayleigh damping formulation, is avoided. Figure 2 illustrates all nonlinear and damping models used in this research explicitly.
5
ACCEPTED MANUSCRIPT
Models Concentrated Plasticity Based On Zareian and Medina's Approach
Distributed plasticity Displacement Beam Column
Beam With Hinge
n=1
n=5
n=10
T
Force Beam Column
CR
IP
πΌ0 π + π½0 πΎππππ‘πππ πΌ0 π + π½0 πΎππππππ‘ πΌ0 π + π½0 πΎππ’πππππ‘ πΌπ‘ π + π½π‘ πΎππ’πππππ‘
US
Fig. 2 Concise illustration of damping and nonlinear models used in this paper
Table 5 Description of nonlinear element types used in this research
Force Beam-Column
Beam With Hinge
Zareian and Medina's Proposed Approach
Displacement BeamColumn
Elastic + Zero length (spring) elements
AC
CE
Concentrated Plasticity
PT
Stiffness Approach
Table 6 Damping formulations associated with distributed plasticity models* πΆ = πΌ0 π + π½0 πΎππππ‘πππ πΆ = πΌ0 π + π½0 πΎπππ π‘ ππππππ‘ππ πΆ = πΌ0 π + π½0 πΎππ’πππππ‘ πΆ = πΌπ‘ π + π½π‘ πΎππ’πππππ‘
definition
AN
Flexibility Approach (Using force interpolation functions [18])
type of element
ED
Distributed Plasticity
Origin of elements' definition
M
Type of inelastic behavior in elements
The total elements' length has the potential of inelastic behavior by definition of integration points and fiber sections. A specified length at two ends of the elements, to which fiber sections are assigned, can have nonlinear behavior; other parts are elastic. The whole length of element can have inelastic behavior, however, elements should be divided into parts due to the stiffness method's assumptions, thereby achieving more reliable results. Inelastic behavior is possible to occur in two springs at two ends of elements to which no damping is assigned The middle part of elements to which modified initial stiffness proportional damping is assigned, has elastic behavior [8].
* Descriptions: πΌ0 ,π½0 : Constant proportionality terms based on the original system. πΎππππ‘πππ : Initial stiffness matrix of the structure πΎπππ π‘ ππππππ‘ππ : The last converged stiffness matrix from the previous step of the analysis defined in OPENSEES program πΎππ’πππππ‘ : Tangent stiffness matrix of the structure πΌπ‘ ,π½π‘ : Updated proportionality terms based on tangent stiffness and current frequency of the structure
6
ACCEPTED MANUSCRIPT
CR
IP
T
The main idea of considering three different values for "n" ratio comes from the contention in the literature on choosing the best value for "n". The best theoretical combination would be a perfectly plastic spring with a very high stiffness connected by an elastic beam representing the whole elastic stiffness. Similarly, in the report written by Medina & Krawinkler [19], n=10 is suggested for moment resisting frames which may be due to the point that using large values for "n" is rational which leads to avoiding any rotation in springs when they are not active. In other words, as long as avoiding ill conditioning, "n" should be large provided that the objective is to ensure that the rotational spring in elastic mode acts as a penalty to ensure continuity in rotation across the joint. On the other hand, in the report written by Zareian and Krawinkler [20], the convergence problems and ill conditioning resulting from considering large values for "n" is mentioned as a disadvantage of this modeling method and it is proposed to use n=1 for generic moment resisting frames. In order to evaluate the effect of considering different values for "n" in concentrated plasticity approach on the fundamental period of RC frames, table 7 including the fundamental period of three frames associated with the models with n=1, n=5, and n=10 for both four and eight story frames is drawn.
AN
US
Table 7 Fundamental periods of concentrated plasticity models Frame n=1 n=5 n=10 Four Story 0.711 0.577 0.542 Eight Story 1.017 0.827 0.777
AC
CE
PT
ED
M
The rationale of conducting a comparative study on such frames with a slight difference in their fundamental period comes from a deep look into the plastic hinge models. In concrete and steel structures, the only reliable studies in this regard are those conducted by Haselton et.al [21] and Lignos and Krawinkler [22] in which they mainly offered a set of regression-based backbone curves coupled with hysteresis rules to explicitly incorporate the cyclic deterioration, all extracted from experimental data. So there is a single nonlinear curve for each element consisting of four linear parts from: 1- origin to the yield point, 2-then to capping point, 3- then to residual strength point, and 4- then to ultimate point. The first part of this multi-linear back-bone curve presents effective elastic stiffness; hence, even using a large "n" will not necessarily result in a fundamental period equal to an elastic model. On the other hand, the whole philosophy of inventing "n" is to formulate an elastic element in a series with plastic hinge which enables us to circumvent the problem of ill condition while its stiffness modifier theoretically keeps the initial elastic stiffness matrix intact. While using different n values, actually it is essential to modify the nonlinear back bone curve to which does not perfectly represent initial stiffness of the structure itself. From another aspect, considering different values for "n" will definitely cause the model to face change in fundamental period because the stiffness of the end springs and thereby the whole model will decrease by considering lower values for "n" and as a result, the fundamental period of the frame will have a greater value (π = 2πβπ/π, T is the fundamental period, m and k are mass and stiffness matrices of the structure, respectively).
7
ACCEPTED MANUSCRIPT 3. Analysis Procedure and Evaluation of Results 3.1 Incremental Dynamic Analysis
T
Owing to the point that performance assessment in the form of loss estimation requires both acceleration and displacement demands, peak floor acceleration (PFA) and peak interstory drift ratio were selected as Engineering Demand Parameters (EDP). First mode spectral acceleration (Sa(T1,5%)) was chosen as Intensity Measure (IM). 22 pairs of ground motion records mentioned in FEMA P695, including longitudinal and transverse records, are used to perform IDA analysis [16]. Table 8 indicates a summary of ground motion records used in this paper.
IP
Table 8 Summary of the far-field record set used in this paper [16] Magnitude
Year
Earthquake Name
Recording Station Name
1
6.7
1994
Northridge
Beverly Hills-Mulhol
2
6.7
1994
Northridge
3
7.1
1999
Duzce, Turkey
4
7.1
1999
Hector Mine
5
6.5
1979
Imperial Vally
Delta
6
6.5
1979
Imperial Vally
El Centro Array #11
7
6.9
1995
Kobe, Japan
Nishi-Akashi
8
6.9
1995
9
7.5
10
CR
ID No.
Canyon Country-WLC Bolu
AN
US
Hector
Shin-Osaka
1999
Kocaeli, Turkey
Duzce
7.5
1999
Kocaeli, Turkey
Arcelik
11
7.3
1992
12
7.3
1992
13
6.9
14
6.9
15
7.4
M
Kobe, Japan
Yermo Fire Station
Landers
Coolwater
1989
Loma Prieta
Capitola
1989
Loma Prieta
Gilroy Array #3
ED
Landers
Abbar
1987
Superstition Hills
El Centro Imp. Co.
6.5
1987
Superstition Hills
Poe Road (temp)
7
1992
Cape Mendocino
Rio Dell Overpass
19
7.6
1999
Chi-Chi, Taiwan
CHY101
20
7.6
1999
Chi-Chi, Taiwan
TCU045
21
6.6
1971
San Fernando
LA-Hollywood Stor
22
6.5
1976
Friuli, Italy
Tolmezzo
17
AC
CE
18
1990
PT
Manjil, Iran
6.5
16
In this research, the effects of vertical floor acceleration are neglected; the "Newmark-Beta" approach is used as the time integrator; and the "Newton" method is considered as the solution algorithm for the timehistory analyses. Figure 3 illustrates median IDA curves of both RC frames. In general, a similar trend is observable among curves of various damping models for both four and eight story distributed plasticity modeled frames. It is clear that in the elastic range of response, all curves associated with each distributed plasticity model correspond to each other. However, when the behavior of the structure begins to become nonlinear, as it was previously proven, Rayleigh damping model with initial stiffness proportional term underestimates both drift ratio and peak floor acceleration. Moreover, curves related to damping models with Kcurrent (tangent stiffness)
8
ACCEPTED MANUSCRIPT
IP
T
are approximately consistent with the curves derived from models with Klastcommited (last converged stiffness matrix gotten from previous step of analysis), considering the point that computational efforts for the former model was much more considerable than the latter one. Therefore, it can be concluded that it is thoroughly safe to use Klastcommited in lieu of Kcurrent in order to reduce run time. It can also be seen that the curves related to approach C of table 1 (using tangent stiffness matrix with updated proportionality terms) tend to be the lowest curves, which means that this model leads frames to higher amounts of demand parameters in a specific IM. As it is evident in the curves associated with concentrated plasticity models in figure 3, the higher the n ratio, the less the response in a particular IM level. In other words, implementation of a flexible spring (n=1) leads the analyst to get the highest estimation of demand parameters among all models. In general, it is obvious that the damping model selection, in line with the choice of the structureβs nonlinear formulation, can have a significant impact on the estimation of responses.
CR
3.2 Collapse Assessment
AC
CE
PT
ED
M
AN
US
In this paper, collapse limit-state is defined as when the tangent of the IDA curve gets 20% of the starting (elastic) slope and lower, peak interstory drift ratio reaches 10%, or dynamic instability occurs [23]. Figure 4 depicts collapse fragility curves which illustrate the probability of collapse with respect to each specific spectral acceleration. In this figure, it is evident that using Rayleigh damping model based on mass and initial stiffness matrix underestimates probability of collapse compared to all distributed plasticity-modeled frames as well as using rigid rotational springs in concentrated plasticity models. This means that these two damping modeling strategies may provide less conservative estimation of collapse capacity. In order to discuss the impact of damping modeling selection on the collapse capacities of all models more conveniently in detail, median collapse capacities equivalent to the spectral accelerations in which the probability of collapse is equal to 0.5, are extracted and represented in figure 5. According to this figure, as it was above-mentioned, the highest level of median collapse capacity belongs to Rayleigh damping model based on mass and initial stiffness proportional damping. This result is congruent with the current practice which argues that such damping model overestimates damping forces and underestimates demand parameters such as interstory drift ratio and peak floor acceleration. Consequently, the structure will be analyzed and designed less conservatively by using such damping model. In plain words, using this damping model overestimates buildingβs collapse capacity. Figure 5 also indicates that the difference between median collapse capacities of distributed plasticity modeled frames with Rayleigh damping using Kcurrent and Klast committed are negligible; hence, Klast committed can safely be used in lieu of Kcurrent for modeling Rayleigh damping in OPENSEES program. It can also reduce computational efforts and prevent convergence difficulties. Another considerable point in figure 5 is that using higher n ratio, which means using rigid rotational springs in models, has resulted in larger estimation of median collapse capacity. In other words, implementation of flexible springs (n=1) is a highly conservative way of evaluating collapse capacity and thereby modeling the structure. Another observable point in figure 5 is that in general, distributed plasticity models lead to higher collapse capacity estimation of frames than concentrated ones. This may be because of the difficulty in simulating deterioration due to local buckling of steel reinforcements and also nonlinear interaction of shear and flexural behavior, thereby neglecting such points in fiber sections. Whereas, local behaviors are more convenient to be simulated through concentrated plasticity formulation [24].
9
ACCEPTED MANUSCRIPT
DispBeamColumn
ForceBeamColumn
6
DispBeamColumn
5
ForceBeamColumn
5
Sa (g)
6
0 0.1
Drift
BeamWithHinge
0.1
Drift
Concentrated Plasticity
4
0
0
0
2.5
PFA (g)
BeamWithHinge
5
4
PFA (g)
2.5
Concentrated Plasticity
0 0
0 0
0.1
Drift
(a) DispBeamColumn
ForceBeamColumn
5
PFA (g)
0
0 0
2.5
PFA (g)
2.5
(b)
DispBeamColumn
ForceBeamColumn
5
5
0
ED
Sa (g)
M
5
0.1
Drift
AN
0
US
Sa (g)
CR
6
0 0
T
0
IP
0
0
0.1
drift
BeamWithHinge
0
PT
0
drift
0.1
0 0
4
BeamWithHinge
Concentrated Plasticity
drift
0 0.1
drift
0.1
(c)
4
3
0 0
PFA (g)
Concentrated Plasticity
5
CE 0
AC
Sa (g) 0
0
PFA (g)
4
5
0
0 0
PFA (g)
4
0
PFA (g)
4
(d)
Distributed plasticity
Concentrated plasticity n=1 n=5 n=10
πΌ0 π + π½0 πΎππππ‘πππ πΌ0 π + π½0 πΎππππππ‘ πΌ0 π + π½0 πΎππ’πππππ‘ πΌπ‘ π + π½π‘ πΎππ’πππππ‘
Fig. 3 Median IDA curves for 4-story frame (a) EDP=drift (b) EDP= PFA; and 8-story frame (c) EDP=drift (d) EDP=PFA
10
ACCEPTED MANUSCRIPT
DispBeamColumn
ForceBeamColumn
1
1
1
DispBeamColumn
ForceBeamColumn
Probability of collapse
1
0
BeamWithHinge
Concentrated Plasticity
1
0
Sa (g)
0
7
7
BeamWithHinge
1
Sa (g)
0
7
Concentrated Plasticity
1
US
CR
Probability of Collapse
1
Sa (g)
0
7
IP
Sa (g)
0
T
0
0
0
Sa (g)
0
7
0
Sa (g)
0
7
(a) Distributed plasticity
0
Sa (g)
0
AN
0
7
0
Sa (g)
7
(b) Concentrated plasticity
πΌ0 π + π½0 πΎππππ‘πππ πΌ0 π + π½0 πΎππππππ‘ πΌ0 π + π½0 πΎππ’πππππ‘ πΌπ‘ π + π½π‘ πΎππ’πππππ‘
ED
M
n=1 n=5 n=10
PT
Fig. 4 Collapse fragility curves for (a) 4-story (b) 8-story frames
3 2 1 0
4story
CE
Sa (g)
4
8story
AC
DispBeamColumn Ξ±M+Ξ²Kinit
4story
8story
4story
ForceBeamColumn
Ξ±M+Ξ²Kcommit
8story
BeamWithHinge
Ξ±M+Ξ²Kcurr
Ξ±tM+Ξ²tKcurr
4story
8story
Concentrated Plasticity n=1
n=5
n=10
Fig. 5 Comparison of median collapse capacities for all models
From another aspect, the comparison of standard deviation of median collapse capacities regarding all four damping models for each nonlinear element type as a measure of variability in terms of both building height and nonlinear elements themselves is discussed. At first, the sensitivity of nonlinear elements to damping modeling method is assessed when the building height increases (comparison between 4 and 8-story frames). As it is indicated in the table 9, among distributed plasticity models, standard deviation of collapse capacities for displacement- beam-column elements has been increased for 8-story frame compared to 4-story one, while this quantity for force-beam-column element is reduced; also, the difference of such quantity between 4 and 8-story frames for beam-with-hinge element is
11
ACCEPTED MANUSCRIPT
T
negligible. This means that displacement-beam-column elements are more sensitive than other distributed plasticity element types to the damping modeling method when the buildingβs height increases. Secondly, standard deviations for one frame in terms of different nonlinear formulations is analyzed. According to table 9, the highest amount of standard deviation belongs to concentrated plasticity models, tending to have a high sensitivity to the n ratio. As a result, adjusting n ratio in this damping modeling technique can have a considerable impact on collapse capacity estimation of the building. Moreover, among distributed plasticity models, elements formulated based on flexibility method have the lowest standard deviation which demonstrates that these element types are less sensitive to the Rayleigh damping modeling strategy.
CR
IP
Table 9 Standard Deviation of median collapse capacities of four damping models element type 4-story 8-story Displacement Beam Column 0.190 0.236 Distributed Force Beam Column 0.191 0.129 Plasticity Beam With Hinge 0.161 0.171 Concentrated Plasticity 0.317 0.279
AC
CE
PT
ED
M
AN
US
Considering the assessment of collapse mechanism in frames modeled this study, it has been observed that each frame exhibits approximately the same collapse mechanism for all 44 ground motions. As a result, four frames as samples, are selected to be depicted under the transverse component of Manjil ground motion record. Figure 6 visually represents the onset of collapse in 4 and 8 story frames with two selected damping models which are based on distributed plasticity (FBC-Rayleigh damping with mass and initial stiffness proportional damping) and concentrated plasticity formulations (n=1). According to figure 6.b and 6.d, concentrated plasticity models (whose sample is chosen to be the model with n=1) commonly show the mechanism of collapse of second and third floors. Occasionally, in some cases, the plastic hinges begin to form in exterior columns, but typically, the mechanism of collapse of floors were observed for concentrated plasticity modeled frames. The typical collapse mechanism of frames simulated by distributed plasticity formulation is observable in figure 6.a and 6.c. Due to similarity of the collapse mechanism among frames with distributed plasticity formulation, the frame with force-beam-column elements and Rayleigh damping based on mass and initial stiffness proportional damping is selected as a representative sample for all 12 frames of this group for each 4 and 8 story frames.
12
IP
T
ACCEPTED MANUSCRIPT
(b)
CE
PT
ED
M
AN
US
CR
(a)
(c)
(d)
AC
Fig. 6 Incipient collapse shape for a) 4 story frame (FBC-Ξ±M+Ξ²Kinit), b) 4 story frame (n=1), c) 8 story frame (FBCΞ±M+Ξ²Kinit) and d) 8 story frame (n=1)
One of the important parameters in performance assessment in order to quantify building collapse risk is annual probability of collapse. It can be derived by means of mean annual frequency of collapse (Ξ»c) which is influential in combination of seismic hazard curve of the building site and fragility curve which culminates in calculating annual probability of collapse [25]. Figure 7 illustrates two hazard curves used in this research for 4 and 8-story frame drawn by means of available database on USGS website [26].
13
Annual Frequency of Exceedance
ACCEPTED MANUSCRIPT 0.04
0.016
0.03
0.012
0.02
0.008
0.01
0.004
0
0 0
0.5
1
1.5
Sa (g)
2
2.5
3
3.5
0
0.5
(a)
1
1.5
Sa (g)
2
2.5
3
(b)
T
Fig. 7 Hazard curves for (a) 4-story (b) 8-story frames
M
3.20E-04 1.60E-04 8.00E-05 0.00E+00
PACT
SP3
DispBeamColumn
PACT
SP3
ForceBeamcolumn
Ξ±M+Ξ²Kcommit
Ξ±M+Ξ²Kcurr
CE
Ξ±M+Ξ²Kinit
ED
2.40E-04
PT
Annual Probablity of Collapse
AN
US
CR
IP
In this research, performance assessment is conducted by two relevant programs: The Performance Assessment Calculation Tool (PACT) which has been introduced in FEMA P58 [27], and Seismic Performance Prediction Program (SP3) introduced by Haselton and Baker Risk Group [28]. Although PACT has been widely used by researchers and has produced acceptable results, SP3 has some advantages over PACT that make it more user-friendly. For instance, the hazard curve is accessible online and the user has to only enter the buildingβs location into the program; and the building components can be defined automatically and conveniently. For the sake of brevity, annual probabilities of collapse derived from both programs are compared for the 4-story frame in figure 8 as an example. The performance assessment results shown in what follows, are the ones which are derived from SP3 program. As it is evident in the figure 8, there is a negligible difference between annual probabilities of collapse derived from the two programs. This may be due to the little difference between the algorithms used by these programs and also because of the point that hazard curves available on SP3 database are more accurate than the hazard curves used in PACT into which have to be entered manually.
PACT
SP3
BeamWithHinge Ξ±tM+Ξ²tKcurr
PACT
SP3
Concentrated Plasticity n=1
n=5
n=10
Fig. 8 Comparison of results derived from PACT and SP3 programs for the 4-story frame
AC
Annual probability of collapse for 4 and 8-story frames are represented in tables 10 and 11 respectively. In order to compare all models conveniently, column chart is drawn in figure 9. According to this figure, selecting n=1 in concentrated plasticity models leads to a considerable increase in annual probability of collapse, i.e. a conservative analysis of the building compared to other models. Additionally, among distributed plasticity models, Rayleigh damping with mass and tangent stiffness matrix coupled with updated proportionality terms shows higher annual probability of collapse. Similar to previous results, Rayleigh damping with mass and initial stiffness proportional damping provides the lowest annual probability of collapse, thereby the least conservative estimate of this parameter.
14
ACCEPTED MANUSCRIPT Table 10 Annual Probability of Collapse for 4-story frame Distributed plasticity Concentrated plasticity
Model
Ξ±M+Ξ²Kinit
Ξ±M+Ξ²Kcommit
Displacement Beam Column Force Beam Column Beam With Hinge n=1 n=5 n=10
0.000052 0.000045 0.000071
0.000081 0.000064 0.000096
Ξ±M+Ξ²Kcurr
0.000081 0.000066 0.000096 0.000277 0.000132 0.000086
Ξ±tM+Ξ²tKcurr 0.000102 0.000092 0.000126
Table 11 Annual Probability of Collapse for 8-story frame
Ξ±M+Ξ²Kcommit
Displacement Beam Column Force Beam Column Beam With Hinge n=1 n=5 n=10
0.000006 0.000014 0.000015
0.000014 0.000023 0.00003
0.0003
IP
T
0.000015 0.000021 0.000027 0.000128 0.000037 0.000034
Ξ±tM+Ξ²tKcurr 0.000018 0.000024 0.000031
Ξ±M+Ξ²Kinit Ξ±M+Ξ²Kcommit Ξ±M+Ξ²Kcurr Ξ±tM+Ξ²tKcurr concentrated
0.00012
0.0002
AN
0.00008
0.0001
0.00004
0
M
0
(a)
ED
Annual Probability of Collapse
Ξ±M+Ξ²Kcurr
US
Concentrated plasticity
Ξ±M+Ξ²Kinit
CR
Distributed plasticity
Model
(b)
PT
Fig. 9 Annual Probability of Collapse for (a) 4-story (b) 8-story frames
3.3 Mean Loss
AC
CE
In this part, the mean repair cost of buildings corresponding to each intensity level is demonstrated. This is obtained through definition of fragility and consequence functions of all building components and summation of their repair costs. If collapse occurs for a building, the repair cost will be commensurate with its determined replacement cost [1]. Figure 10 illustrates mean repair cost curves for both frames. According to figure 10, among all distributed plasticity models, mass and initial stiffness proportional damping model has brought about the least estimation of normalized mean repair cost in a specific intensity level; while, Rayleigh damping based on mass and tangent stiffness matrices with updated proportionality terms has the highest evaluation of such quantity. Consequently, using the latter damping model leads to more conservative judgment of building repair cost than the former.
15
DispBeamColumn
25
DispBeamColumn
50
20
40
40
15
15
30
30
10
10
20
20
5
5
10
10
0
BeamWithHinge
25
0 0
3.5
3.5
Concentrated Plasticity
25
20
1.75
15
10
10
5
5
1.4
1.75 Sa (g)
3.5
0 2
0
3
1.4 Sa (g)
Sa (g)
US
(a)
10
0
2.8
0
1.2 Sa (g)
2.4
(b)
Concentrated plasticity n=1 n=5 n=10
M
πΌ0 π + π½0 πΎππππ‘πππ πΌ0 π + π½0 πΎππππππ‘ πΌ0 π + π½0 πΎππ’πππππ‘ πΌπ‘ π + π½π‘ πΎππ’πππππ‘
AN
Distributed plasticity
Concentrated Plasticity
20
10 1
2.8
30
20
0
1.4
40
40
0 0
0 50
50
30
0
2.8
BeamWithHinge
60
20
15
0 0
T
1.75
IP
0
ForceBeamColumn
50
20
0 Mean Loss Normalized by Total Replacement Cost (%)
ForceBeamColumn
25
CR
Mean Loss Normalized by Total Replacement Cost (%)
ACCEPTED MANUSCRIPT
Fig. 10 Mean repair cost normalized by total replacement cost for (a) 4-story (b) 8-story frames
CE
PT
ED
Moreover, in the 4-story frame, implementation of various n ratios has led to more varied quantities than the 8-story one. As it is evident in the figure 10.b, curves corresponding to n=5 and n=10 are approximately consistent for the 8-story frame which indicates that using rigid and semi-rigid rotational springs can result in a same evaluation of building repair cost for the 8-story frame. From another aspect, for the 4-story frame, the difference between normalized mean repair costs of n=1 and n=10 at the intensity level equal to 2.65g is roughly 78% because the model with n=1 has collapsed at Sa=2.15g and the repair cost at Sa=2.65g is equal to 100%. Similarly, the same trend is observable at Sa=2.15g for the 8-story frame in figure 10.b. Therefore, using a flexible rotational spring, the analyst will obtain a considerably conservative evaluation of repair cost.
AC
3.4 Expected Annual Loss
The amount of money on average which has to be paid annually to repair building damages likely to occur due to the probable earthquakes is equal to the expected annual loss (EAL). According to Dhakal and Mander [29], this quantity can be measured by integrating the normalized repair cost over the possible hazard frequencies. Table 12 Expected Annual Loss for 4-story frame ($) Distributed plasticity Concentrated plasticity
Model
Ξ±M+Ξ²Kinit
Ξ±M+Ξ²Kcommit
Ξ±M+Ξ²Kcurr
Ξ±tM+Ξ²tKcurr
Displacement Beam Column Force Beam Column Beam With Hinge n=1 n=5 n=10
24341 23563 25326
30630 27513 26853
30560 27671 27076
33417 30986 28743
49189 27417 21686
16
ACCEPTED MANUSCRIPT
Table 13 Expected Annual Loss for 8-story frame ($)
Ξ±M+Ξ²Kcommit
Ξ±M+Ξ²Kcurr
Ξ±tM+Ξ²tKcurr
23186 23721 21363
28548 27304 24273
28842 27431 25315
29987 27900 25673
41827 20848 17589 45000
55000 44000
30000
33000 22000
T
EAL ($)
Concentrated plasticity
Ξ±M+Ξ²Kinit
Displacement Beam Column Force Beam Column Beam With Hinge n=1 n=5 n=10
15000
11000 0
(b)
AN
Fig. 11 Expected Annual Loss for (a) 4-story (b) 8-story frame
US
CR
0
(a)
Ξ±M+Ξ²Kinit Ξ±M+Ξ²Kcommit Ξ±M+Ξ²Kcurr Ξ±tM+Ξ²tKcurr concentrated
IP
Distributed plasticity
Model
ED
M
As it is apparent in tables 12 and 13 as well as figure 11, concentrated plasticity models have the highest and lowest estimation of EAL. It can be concluded that a flexible rotational spring implementation in the model (i.e. n=1) can lead to a highly conservative evaluation of EAL; which is congruent with the assessment of normalized mean losses as well. Moreover, among distributed plasticity models, mass and initial stiffness proportional damping model provides less conservative results compared to other damping models. Table 14, similar to table 9, is drawn to indicate the standard deviations associated with the derived EAL values in order to quantify variation in this parameter.
CE
PT
Table 14 Standard Deviation of expected annual loss of four damping models element type 4-story 8-story Displacement Beam Column 3322 2628 Distributed Force Beam Column 2630 1671 Plasticity Beam With Hinge 1211 1693 Concentrated Plasticity 11848 10740
AC
Considering table 14, it can be seen that the trend observed for this parameter is similar to the trend observed in the median collapse capacity parameter indicated in table 8. However, since the expected annual loss seems to have much more varied values and much sensitive to different damping models, the difference between the standard deviations is much higher. It is evident that concentrated plasticity models have provided more divergent values than distributed plasticity models.
4. Conclusions In this study, the effects of various and currently used damping modeling techniques coupled with implementation of different nonlinear elements on the seismic response and performance parameters are investigated. Distributed plasticity formulation was used, comprising 12 nonlinear elements (displacementbased and force-based) with different Rayleigh damping modeling strategies. Additionally, 3 concentrated plasticity models were developed, including rigid, semi-rigid, and flexible rotational end springs which have no allocated damping and with the modified stiffness proportional damping assigned to the elastic parts of the 17
ACCEPTED MANUSCRIPT elements. These models were applied to 4 and 8-story RC moment frames. Seismic responses of two frames were evaluated by incremental dynamic analysis, then, collapse and performance assessments were carried out in accordance with FEMA P58 by calculating median collapse capacity, annual probability of collapse, mean repair cost, and expected annual loss. The results derived from this research are elaborated in what follows:
T
PT
AC
CE
ο·
ED
M
ο·
AN
ο·
US
CR
ο·
Using Rayleigh damping formulation based on mass and initial stiffness matrices may result in overestimation of collapse capacity and underestimation of performance parameters. This can be considered as a consequence of overestimation of damping forces and underestimation of demand parameters such as story drift by using this damping model which have been presented by researchers hitherto. Among distributed plasticity models, using Rayleigh damping based on mass and tangent stiffness matrices with updated proportionality terms may lead to a higher estimation of building response, lower estimation of collapse capacity, and higher evaluation of performance parameters, leading the analyst to reach a conservative estimate of such quantities. However, it should be noted that due to the necessity for recalculating frequencies and thereby proportionality terms in each step of analysis and utilizing tangent stiffness matrix, this damping modeling method is inefficiently time-consuming. The results associated with Rayleigh damping based on mass and tangent stiffness matrix (Kcurrent) and last committed stiffness (Kcommit) show an approximate consistency. Therefore, considering convergence issues and prolongation of analysis caused by using Kcurrent, it may be judicious to apply Kcommit in lieu of Kcurrent which results in more time-saving method. Concentrated plasticity models represent a high sensitivity to adjusting n ratio (i.e. stiffness of rotational end springs). The comparison between results derived from using various n ratios in concentrated plasticity models demonstrates that the implementation of a flexible spring may lead the analyst to a greatly conservative estimation of performance parameters; therefore, the stakeholder may make an inordinate investment. On the other hand, using a rigid spring brings about higher evaluation of collapse capacity and lower estimation of performance parameters than other concentrated plasticity models. In this paper, two performance assessment programs were used: PACT and SP3. All results obtained from both programs were roughly equivalent. However, SP3 program is more user-friendly because the user is not required to enter nonlinear dynamic analysis results manually (it is possible by uploading an excel file), the building components can be added to the model automatically, etc. Moreover, it seems to be more accurate than PACT because it has the direct access to the USGS hazard database.
IP
ο·
It is significant to notice that this paper is a comparative analytical study considering numerical and analytical approaches for all models of four and eight story RC frames and should be interpreted with caution. Due to the scarcity of experimental data about this topic in the literature, focusing on inherent damping modeling methods and evaluating their effects on the presently widely used nonlinear models have to be performed by numerical and analytical comparative studies. Yet, sophisticated experimental studies on damping modeling methods and their effect on structural systems' response are highly needed. This would enhance the reliability of analytical studies and possibly lead to the determination of the best damping modeling technique.
References [1]
FEMA P58- Federal Emergency Management Agency. Seismic Performance Assessment of Buildings Volume 1- methodology. Appl Technol Counc 2012;1:278. 18
ACCEPTED MANUSCRIPT Rayleigh, Lord. The Theory of Sound, Vol 1. New York: Dover Publications; 1896.
[3]
Charney F a. Unintended Consequences of Modeling Damping in Structures. J Struct Eng 2008;134:581β92. doi:10.1061/(ASCE)0733-9445(2008)134:4(581).
[4]
Hall JF. Problems encountered from the use (or misuse) of Rayleigh damping. Earthq Eng Struct Dyn 2006;35:525β45. doi:10.1002/eqe.541.
[5]
Bernal D. Viscous Damping in Inelastic Structural Response. J Struct Eng 1994;120:1240β54.
[6]
Petrini L, Maggi C, Priestley MJN, Calvi GM. Experimental Verification of Viscous Damping Modeling for Inelastic Time History Analyzes. J Earthq Eng 2008;12:125β45. doi:10.1080/13632460801925822.
[7]
Erduran E. Measuring bias in structural response caused by ground motion scaling. Earthq Eng Struct Dyn 2012;41:1905β19. doi:10.1002/eqe2164.
[8]
Zareian F, Medina RA. A practical method for proper modeling of structural damping in inelastic plane structural systems. Comput Struct 2010;88:45β53. doi:10.1016/j.compstruc.2009.08.001.
[9]
Jehel P, Leger P, Ibrahimbegovic A. Initial versus tangent stiffness-based Rayleigh damping in inelastic time history seismic analyses. Earthq Eng Struct Dyn 2014;43:467β84. doi:10.1002/eqe.2357.
[10]
Chopra AK, Mckenna F. Modeling viscous damping in nonlinear response history analysis of buildings for earthquake excitation. Earthq Eng Struct Dyn 2015;45:193β211. doi:10.1002/eqe.2622.
[11]
Hardyniec A, Charney F. An investigation into the effects of damping and nonlinear geometry models in earthquake engineering analysis. Earthq Eng Struct Dyn 2015;44:2695β715. doi:10.1002/eqe.2604.
[12]
Puthanpurayil AM, Lavan O, Carr AJ, Dhakal RP. Elemental damping formulation: an alternative modelling of inherent damping in nonlinear dynamic analysis. Bull Earthq Eng 2016;14:2405β34. doi:10.1007/s10518016-9904-9.
[13]
Luco JE, Lanzi A. Optimal Caughey series representation of classical damping matrices. Soil Dyn Earthq Eng 2017;92:253β65. doi:10.1016/j.soildyn.2016.10.028.
[14]
Luco JE, Lanzi A. A new inherent damping model for inelastic time-history analyses. Earthq Eng Struct Dyn 2017;46:1919β39. doi:10.1002/eqe.2886.
[15]
OPENSEES. Open System for Earthquake Engineering Simulation. Pacific Earthq Eng Res Center, Univ Calif Berkeley, CA 2002. http://opensees.berkeley.edu/ (accessed March 17, 2016).
[16]
FEMA P695- Federal Emergency Management Agency. Quantification of building seismic performance factors. Appl Technol Counc 2009:421.
[17]
Mander JB, Priestley MJN, R.Park. Theoretical Stress Strain Model for Confined Concrete. J Struct Eng 1988;114:1804β25. doi:10.1061/(ASCE)0733-9445(1988)114:8(1804).
[18]
Neuenhofer A, Filippou F. Evaluation of Nonlinear Frame Finite-Element Models. J Struct Eng 1997;123:958, 966. doi:10.1061/(ASCE)0733-9445(1997)123.
[19]
Medina R, Krawinkler H. Seismic Demands for Nondeteriorating Frame Structures and Their Dependence on Ground Motions. Pacific Earthq Eng Res Center, Univ Calif Berkeley, CA 2004.
[20]
Zareian F, Krawinkler H. Simplified performance-based earthquake engineering. John A Blume Earthq Eng Res Cent 2009:338.
[21]
Haselton CB, Liel AB, Lange ST. Beam-Column Element Model Calibrated for Predicting Flexural Response Leading to Global Collapse of RC Frame Buildings. Report No.03, Pacific Earthq Eng Res Center, Univ Calif Berkeley, CA. 2008.
[22]
Lignos DG, Krawinkler H. Sidesway Collapse of Deteriorating Structural Systems under Seismic Excitations. John A Blume Earthq Eng Res Cent 2009.
[23]
Vamvatsikos D, Cornell CA. The Incremental Dynamic Analysis and Its Application To Performance-Based Earthquake Engineering. Eur Conf Earthq Eng 2002:10. doi:10.1002/eqe.141.
AC
CE
PT
ED
M
AN
US
CR
IP
T
[2]
19
ACCEPTED MANUSCRIPT Deierlein GG, Reinhorn AM, Willford MR. Nonlinear Structural Analysis For Seismic Design: A Guide for Practicing Engineers. NEHRP Seism Des Tech Br No 4 2010:36.
[25]
Eads L, Miranda E, Krawinkler H, Lignos DG. Improved Estimation of Collapse Risk for Structures in Seismic Regions. FIFTHTEENTH WORLD Conf Earthq Eng 2012.
[26]
USGS. United States Geological Survey n.d. https://earthquake.usgs.gov/hazards/ (accessed June 12, 2016).
[27]
FEMA P58- Federal Emergency Management Agency. Seismic Performance Assessment of Buildings Volume 2 - Implementation Guide. Appl Technol Counc 2012;2:357.
[28]
Seismic Performance Prediction Program by Haselton Baker Risk group. SP3 2015. https://www.hbrisk.com/ (accessed October 23, 2016).
[29]
Dhakal RP, Mander JB. Financial Risk Assessment Methodology for Natural Hazards. Bull New Zeal Soc Earthq Eng 2006:91β105.
AC
CE
PT
ED
M
AN
US
CR
IP
T
[24]
20