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Floor response spectra for beyond design basis seismic demand
Nuclear Engineering and Design xxx (2017) xxx–xxx
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Floor response spectra for beyond design basis seismic demand Somnath Jha ⇑, A.D. Roshan, L.R. Bishnoi Siting & Structural Engineering Division, Atomic Energy Regulatory Board, Mumbai 400094, India
a r t i c l e
i n f o
Article history: Received 31 May 2016 Received in revised form 23 December 2016 Accepted 4 January 2017 Available online xxxx
a b s t r a c t Vulnerability assessment of Structures Systems and Components (SSCs) of nuclear facilities for earthquake ground motion exceeding the design basis ground motion has become a key issue to ensure safety in case of highly improbable but possible extreme earthquake event beyond design basis. In order to assess seismic safety of SSCs located at different floors of the building, floor response spectra (FRS) for such beyond design basis seismic demand, taking into account possible nonlinear behavior of structure, is required. Currently used methods (e.g. IAEA Safety Series 28) do not capture softening of structure at increased demand. A performance based approach would be ideal under such situations. FRS for higher seismic demand may be developed through nonlinear time history analysis of the structure. However, this approach is complicated and cumbersome for structural configurations such as that of NPPs and requires deep insight of solution algorithm and large resources for numerical solution. In this work a simplified methodology for FRS generation for higher level of seismic demand using linear time history analysis of an equivalent linear structural model is proposed. Nonlinear behavior of structure and associated damping are estimated from Nonlinear Static Pushover Analysis. Through this approach realistic estimation of stiffness degradation of the structure and increase in damping at different levels of seismic demands can be made. Subsequently, linear time history analysis is performed on degraded structure, accounting for stiffness reduction and hysteretic damping to obtain the floor time histories which are then converted to FRS. Application of this method is shown through case studies. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Seismic safety is one of the essential design requirements of a NPP structure. Occurrence of recent seismic events (NCO earthquake of 2007 and GEJE event of 2011) that resulted in ground motions beyond design basis at many NPP sites has emphasized the need for assessment of margins to ensure safety against beyond design basis events. With increase in demand, structure may exhibit nonlinear behavior resulting in softening (stiffness reduction) and change in natural frequencies. Nonlinear behavior of the structure also results in additional damping due to hysteresis. Conventional approaches (e.g. IAEA, 2003) address this as an
Abbreviations: ADRS, Acceleration Displacement Response Spectrum; BDBE, Beyond Design Basis Earthquake; CDP, Concrete Damaged Plasticity; CSM, Capacity Spectrum Method; DBE, Design Basis Earthquake; FE, Finite element; FRS, floor response spectra; GEJE, Great East Japan Earthquake; MCE, Maximum Credible Earthquake; NPP, Nuclear Power Plant; NLSPA, Nonlinear Static Pushover Analysis; PGA, Peak Ground Acceleration; SSCs, Structures Systems & Components; SSE, Safe Shutdown Earthquake. ⇑ Corresponding author. E-mail address: [email protected] (S. Jha).
extension of existing seismic analysis methodologies albeit with use of increased damping and response reduction factors due to ductility. Performance assurance of floor mounted equipment and components is important for NPPs even under Beyond Design Basis Earthquake events. Seismic performance of these systems can only be assured if seismic demands on floors is appropriately quantified. Seismic demands on floors is generally represented through Floor response spectra (FRS). FRS for Design Basis Earthquake is developed by linear time history analysis of the structure. Under BDBE, when structure is expected to exhibit inelastic behavior, ideally non-linear time history analysis would be required. Though non-linear time history analysis is a more rational method, its complexity of response tracking, uncertainties associated with material constitutive laws and their impact on solution accuracy are the deterrents for design office use of nonlinear time history analysis. Generally, only specific purpose software like OpenSees (Gregory et al., 2007), IDARC (Valles et al., 1996) can simulate the complex behavior of concrete under cyclic loading. In addition, use of any numerical system to Nuclear Power Plants calls for stringent software related quality assurance (QA) requirements.
http://dx.doi.org/10.1016/j.nucengdes.2017.01.006 0029-5493/Ó 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: Jha, S., et al. Floor response spectra for beyond design basis seismic demand. Nucl. Eng. Des. (2017), http://dx.doi.org/ 10.1016/j.nucengdes.2017.01.006
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Symbols be b bh b0 ED ESo fs f
effective damping viscous damping of shear wall structure pinched hysteresis damping hysteresis damping energy dissipated by damping maximum strain energy secant frequency elastic frequency
Different simplified approaches, as via media, are proposed in the literature for estimation of seismic demands on floors (Igusa and Der Kiureghian, 1985; Villaverde, 1997; Miranda and Taghavi, 2006; CEN EC8, 1998) among others). Limitations of these methods have been extensively reviewed by Sullivan et al. (2013). It is noted that, most approaches in the literature do not consider the impact of different levels of elastic damping and non-linear behavior of the primary structure on floor spectra. These approaches are either simple but not reliable or reliable but require relatively advanced analysis capabilities (Calvi and Sullivan, 2014). Calvi and Sullivan (2014) proposed a simplified method for generation of FRS and validated it by comparing predicted spectra with those obtained from time-history analyses of a case study building. The method proposed in Calvi and Sullivan (2014) could be applied to multi-story structures only in linear range, but it is possible to adjust elastic damping values over a wide range. Development of a simplified approach, which is capable of addressing the non-linear behavior of the primary structure and increased damping that influences the shape and intensity of floor spectra, is quite challenging. The methodology proposed in this work for generation of FRS utilizes state of the art seismic performance assessment procedures viz. Nonlinear Static Pushover Analysis (NLSPA) for accounting non-linear behavior of the structure. Simplicity of pushover analysis makes it more appealing for design office use to calculate degraded stiffness of the structure at different levels of seismic demand exceeding design basis. Hysteretic damping due to nonlinear behavior of the structure could also be evaluated from pushover curve using empirical approaches given in various international guidelines (ATC-40, 1996; Kennedy et al., 1984). The main postulation of the proposed method is that linear time history analysis on equivalent linear structure, after accounting for stiffness degradation and enhanced hysteretic damping can yield floor response spectra for seismic demand, corresponding to Beyond Design Basis Earthquake (BDBE). Industrial standard software platform such as ANSYS (2015) and Abaqus (2010) are being used in nuclear applications after satisfying software quality assurance requirements. The implementation of proposed methodology is shown through the use of Abaqus software. 2. Methodology for generation of FRS using pushover analysis NPP structures may exhibit nonlinear behavior if subjected to ground motion higher than the design basis. FRS required for seismic qualification of floor mounted equipment and components for beyond design basis ground motion should be developed taking into account possible degradation of the structure and expected level of damping. Static pushover analysis is a tool by which stiffness degradation of the structure and expected damping at different levels of seismic demand can be evaluated.
fe KS KI Z I R SRF
effective frequency secant stiffness initial stiffness zone factor (0.24 for Seismic Zone-IV of IS 1893 (2002)) importance factor (1.0 for conventional structure) response reduction factor (5.0 for Shear walls) stiffness reduction factor
In this work, pushover analysis is proposed to be used to determine stiffness degradation of structure and modified damping at different levels of seismic demand. This is done by seismic performance assessment of the structure using Capacity Spectrum Method of ATC-40 (1996). Finite element (FE) model of structure is then modified to take into account the stiffness degradation and modified damping expected at any specified level of seismic motion beyond design basis. Modified FE model of the structure may be named as equivalent linear model. Linear time history analysis of the degraded structure (reduced stiffness) is performed for the modified damping to generate FRS. 2.1. Seismic performance assessment In Capacity Spectrum Method, capacity of structure is compared with seismic demand in Acceleration Displacement Response Spectrum (ADRS) format to assess the seismic performance of structure for a given seismic motion. The procedure seeks to find out a point on the capacity spectrum that also lies on the appropriate demand spectrum, scaled down for nonlinear effects. This point is called performance point. Hysteretic damping and stiffness degradation for the ground motion under consideration is estimated in correspondence of the evaluated performance point. 2.2. Estimation of hysteretic damping Damping that occurs when the earthquake ground motion derives the structure into inelastic range can be viewed as the combination of viscous damping that is inherent in the structure and the hysteretic damping. Hysteretic damping is dependent on area of hysteretic loops formed when the earthquake force (base shear) is plotted against the structure displacement. Hysteretic damping can be represented by equivalent viscous damping using equations available in the literature (ATC-40, 1996; Kennedy et al., 1984; Gülkan et al., 2005). An approximate effective (hysteretic) damping is calculated based on the capacity curve, the estimated displacement demand and resulting hysteresis loop. Probable imperfections in real building hysteresis loops, including degradation and duration effects, are accounted by reducing theoretically calculated equivalent viscous damping values. Equivalent viscous damping of frame structures (i.e. with beam and columns), can be estimated using bilinear hysteretic model given in ATC-40 (1996) and reproduced as Eq. (1). Based on studies of nonlinear response of short period structures, Gülkan et al. (2005) reported that equation given in ATC-40 significantly underestimates the displacement demand of short period structures such as shear walls due to overestimation of viscous damping. High damping values are not expected for shear wall structures due to considerably lower ultimate displacement demand than frame structures and existence of pinching in the hysteresis loop. For experiments conducted on shake table under significant damage
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conditions, the maximum value of damping in RC wall-frame system ranges from 6.9% to 7.5% (PEER/ATC-72-1, 2010), whereas damping predicted by equation given in ATC-40 results in values as high as 20% in some cases. Recognizing this problem, some researchers have proposed equations for estimation of equivalent viscous damping of shear wall structures. Jha et al. (2015) performed a study on prediction of equivalent viscous damping of shear walls by different formulations available in literature. Outcome of the study shows that formulation proposed by Kennedy et al. (1984) (reproduced as Eq. (3)) accounts for structure specific characteristics of shear walls. Thus this formulation was selected for current study. More details on comparison of damping from different formulations and validation of equation used in the current study are available in Jha et al. (2015). 2.2.1. Damping equation as per ATC-40
be ¼ ðb0 þ 0:05Þ
ð1Þ
where
b0 ¼
1 ED 2ðl 1Þð1 aÞ ¼k 4p ESo plð1 þ al aÞ
ð2Þ
wherel ¼ Displacement ductility ratio, a = post elastic stiffness ratio, and k = factor to account structure type. 2.2.2. Damping equation as per Kennedy et al. (1984)
be ¼
fs f fe f
!2 ðb þ bh Þ
ð3Þ
wherebe ¼ effective damping, b ¼ viscous damping of shear wall structure, bh ¼ pinched hysteresis damping, f s ¼ secant frequency, f ¼ elastic frequency, f e ¼ effective frequency.
f bh ¼ 0:11 1 s f fs ¼ f
ð4Þ
rffiffiffiffiffi Ks K
ð5Þ
where K and KS are initial and secant stiffness obtained from pushover curve (See Fig. 1).
fe f ¼ ð1 AÞ þ A s f f
ð6Þ
where A ¼ C F ð1 ffs Þ 6 0:85; &C F ¼ 2:3
Tangent Sffness, K T
Details of formulation to estimate these parameters can be found in ATC-40 and Kennedy et al. (1984). 2.3. Estimation of stiffness degradation Capacity curve obtained using pushover analysis represents degradation of structural stiffness under lateral (seismic) load. Performance point obtained using well established procedures such as Capacity Spectrum Method indicates the maximum level of degradation under seismic demand imposed on the structure. Hence, actual stiffness of the structure at performance point is represented using tangent drawn on the pushover curve at performance point. This is pictorially shown in Fig. 1. It is worth noting that the structure has reached at this point by progressive damage under seismic loading. To derive stiffness degradation from capacity curve to develop equivalent linear model, this progressive degradation of the structure needs to be considered. Equivalent linear methods are most commonly used in Soil Dynamics and Geotechnical Earthquake Engineering for ‘‘site response analysis” in which the free field acceleration of a site with stratified soil deposit is computed using earthquake induced motion at the bedrock. Soil layers subjected to ground motion undergo nonlinear deformations, which needs to be account for their nonlinear behavior. Although ‘‘site response analysis” can be performed using rigorous nonlinear step-by-step time history analysis, this is not the approach followed in practical applications, owing to various uncertainties and computational complexities. The limitations are similar to performing nonlinear time history analysis on reinforced concrete structures. As a general and widely accepted practice, ‘‘equivalent linear method” originally proposed by Seed and Idriss (1970) and implemented in the well-known programs such as SHAKE (Schnabel et al., 1972) is used to calculate the seismic response of the soil deposits. Nonlinear behavior of the soil is approximated by using equivalent shear modulus and damping ratios that are function of effective shear strains. The effective strain is defined by reducing the maximum strain response. A reduction factor of 0.65 is used over the maximum strain value to account for progressive degradation. Feasibility of applying concept of equivalent linear method used in Geotechnical Earthquake Engineering for the present work is assessed. In order to account for progressive degradation of the structure till the performance point, a factor of 1/0.65, is applied on to the degraded stiffness. In other words, to evaluate stiffness of equivalent linear model, tangent stiffness at performance point is increased by a ratio, 1/0.65 (see Fig. 1), to account for the fact that the structural stiffness has degraded to tangent stiffness from initial stiffness, progressively. Thus stiffness reduction factor (SRF) can be obtained by comparing the equivalent stiffness with initial stiffness of the structure. Equation to evaluate SRF is given below.
SRF ¼
Performance point Equivalent Sffness, K T/0.65
Inial Sffness, K I
Fig. 1. Representation of equivalent stiffness.
3
KT 0:65 K I
ð7Þ
This reduction factor is applied to stiffness of all the structural components that contributed to the nonlinear behavior under the specified seismic demand such that the overall structural stiffness is representative of the equivalent softened structure at performance point. This methodology has limitations as the stiffness reduction is not implimented locally to individual members and does not capture progressive softening. However, this approximation would still be reasonable in view of the fact that any local reduction of stiffness would result in redistribution of demand among individual members and thus degrading the global behavior.
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3. Validation of the proposed methodology The proposed methodology is validated for a reinforced concrete structure by comparing experimental results of shake table experiments carried out in the Saclay Nuclear Center in France in 1997 (Locci and et al., 1998). Specimen CAMUS was a 1/3-scale model of a representative 5-story reinforced concrete building detailed according to French practice (IAEA CRP-NFE, 2002). CAMUS specimens were designed considering it as a typical example for a stiff structure commonly used in NPPs. It had two parallel shear walls linked by 6 square floors (1.7 m 1.7 m) (including the floor connected to the footing). The specimen had a total mass of 36 tons with additional masses attached to it. A heavily reinforced concrete footing allowed anchorage to the shaking table. The total height of the model was 5.10 m. Shear walls had a width of 1.7 m and thickness of 6 cm. The stiffness and the strength of the structure in direction perpendicular to shear walls are increased by adding some lateral triangular bracing. The specimen had a measured fundamental frequency of 7.24 Hz and the first vertical eigen mode had a natural frequency of 20 Hz. The dimensions of the specimen are shown in Fig. 2. Other details including seismic weights on floors, reinforcement detailing and material properties etc. can be found in IAEA CRP-NFE (2002). The structure was tested on shake table with mono-axial seismic excitations (parallel to shear wall) with increasing level of input motions. Result of first applied input motion with PGA of 0.25 g is considered for validation in present study. Time history and response spectra of the input motion considered in the experiment are shown in Fig. 3. 3.1. Finite element modeling A 3-D model of the structure is developed in finite element software Abaqus. The actual material properties and boundary conditions in the experiment were implemented in the model to reflect the required aspects of the test specimen. Finite element
model of the structure is shown in Fig. 4. Shear walls, footing and slabs are modeled using 4-noded shell elements (S4R). Triangular bracing provided to prevent non-symmetric failure is modeled using 2-noded beam elements (B31). Average size of finite elements for shear walls was around 0.1 m. This number was arrived at after performing mesh sensitivity studies. Nonlinear behavior of concrete is modeled using Concrete Damaged Plasticity (CDP) material model of Abaqus. This material model has been calibrated and validated for prediction of inelastic behavior of shear wall under lateral loads by comparing experimental data obtained from test conducted on shear walls with varying aspect ratio (Jha et al., 2013). Park et al. (1982) model is used for stress strain curve of concrete under compression and the model proposed by Wahalathantri et al. (2011) is used for tension stiffening. Elasto-plastic model with actual yield and ultimate strength (from experiment) is used for reinforcing steel. 3.2. Pushover analysis of the structure Pushover analysis of the structure is performed using load proportional to fundamental mode. Natural frequency of the structure is calculated to be 7.245 Hz which compares closely with the experimentally observed frequency of 7.24 Hz. Frequency of first vertical mode is calculated to be 20.04 Hz as against measured frequency of 20 Hz. Pushover curve of the structure up to top deflection of 10 mm is shown in Fig. 5. 3.3. Seismic performance assessment Performance point of the structure is evaluated using Capacity Spectrum Method. Hysteretic damping for performance assessment is calculated using Eq. (3) (see Section 2.2.2). Comparison of capacity and demand in ADRS format is shown in Fig. 6. Maximum roof deflection of 7 mm at a base shear of 140 kN was obtained in the experiment. Seismic performance assessment in present study predicted a maximum roof drift of around 6.4 mm
Fig. 2. Camus Specimen (IAEA CRP-NFE, 2002).
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1.2
0.3
1
Sa, g
Acceleraon, g
0.2 0.1 0 0
10
20
30
-0.1
40
0.8 0.6 0.4
Time, s 0.2
-0.2
0
-0.3 0.1
1
10
100
Frequency, Hz Fig. 3. Time history and response spectra of input motion.
Spectral Acceleraon, Sa
10 7.5 5 2.5 0 0
0.0025 0.005 0.0075 Spectral Displacement, Sd
0.01
Fig. 6. Comparison of capacity and demand in ADRS format for CAMUS specimen. Fig. 4. Finite element model of CAMUS specimen. Table 1 Calculation of hysteretic damping for CAMUS specimen.
Base Shear, kN
200 160
Ks K
fs f
A
fe f
be (%)
0.6
0.77
0.53
0.88
5.78
120 80 Table 2 Calculation of stiffness reduction factor (SRF) for CAMUS specimen.
40 0 0
2
4 6 Deflecon, mm
8
10
Fig. 5. Pushover curve of CAMUS specimen.
at a base shear of 155 kN. Thus, performance point of the structure could be predicted with reasonable accuracy. 3.4. Estimation of hysteretic damping and stiffness reduction factor 3.4.1. Hysteretic damping Calculation for hysteretic damping using Eq. (3) at performance point is given in Table 1. 3.4.2. Stiffness reduction factor Calculation for stiffness reduction factor at performance point is given in Table 2.
KT (kN/m)
KI (kN/m)
SRF
1.19E + 04
4.09E + 04
0.446
3.5. FRS using proposed methodology As per the proposed methodology, equivalent linear model of the structure is developed using stiffness reduction factor and hysteretic damping. A linear modal time history analysis is performed for the input motion of the shake table shown in Fig. 3 and corresponding FRS is generated for different floors of the structure. The result is then compared with the FRS obtained from floor time histories recorded during the experiment. As shown in Figs. 7 and 8, FRS obtained using the proposed methodology based on equivalent linear model for different floors are in good agreement with the experimental FRS. The minor differences in some local peaks (Fig. 7a and b) are attributed to the linearization of the non-linear response. The dif-
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1.2
2 Experimental
Equivalent linear analysis Experimental
1
1.6
Equivalent Linear Analysis
0.8
Sa, g
Sa, g
1.2 0.6
0.8
0.4 0.4
0.2 0 0.1
0 1
10
100
Frequency, Hz
0.1
1
(a) 2nd Floor
10
Frequency, Hz
100
(b) 3rd Floor
Fig. 7. Comparison of FRS from experiment and analysis for 2nd and 3rd floor.
6 Experiemental 5 Equivalent linear analysis
Sa, g
4 3 2 1 0 0.1
1
Frequency, Hz
10
100
Fig. 8. Comparison of FRS from experiment and analysis for 6th (top) floor.
ference in the peak value for the top floor (Fig. 8) needs further investigation. 4. Case studies for generation of FRS Owing to large loads and / or shielding considerations, many of the NPP structures are characterized by shear wall type systems. To enable assessment of systems with such configurations, the methodology proposed in this work for generation of FRS under conditions beyond design basis is applied to two sample RCC structures with shear walls. Impact of change in aspect ratio of the wall is also captured in the assessment. 4.1. General details of sample structures Structures considered for the case study consists of two full scale 5 and 3-story buildings with shear walls as main load resisting members. Both the structures have plan dimensions of 24 m 24 m, grid size of 6 m 6 m and floor height of 4 m. Four shear walls are located on the periphery having width of 4 m and 6 m respectively for structure-1 & 2. Indian standard code (2002) has been used for design of these structures. This standard specifies two levels of earthquakes, one is called as Design Basis Earthquake (DBE) and another as maximum considered earthquake. Four seismic zones are defined in this standard starting from
Zone-II (lowest seismicity) to Zone-V (highest seismicity). First structure is designed for Design Basis Earthquake and assumed to be located in a high seismic zone (Zone-IV). Response reduction factor, R, is taken as 5.0 for structure-1. The second structure, three storied RC shear wall-frame building, is considered to be located in same seismic zone but designed for maximum considered earthquake (MCE), which has PGA twice of DBE. Response reduction factor for the second structure is taken as unity in line with principle followed for NPP structures. These structures are designed using two different approaches (different response reduction factor) to demonstrate versatility of the proposed method for different level of damage. Structure-1, designed for lower seismic loads, is expected to suffer more damage when subjected to same level of seismic motion. PGA for Design Basis Earthquake (DBE) is evaluated using following equation given in Indian standard code (2002):
PGA ¼
Z I 2 R
ð8Þ
where Z = seismic zone factor (0.24 for Zone-IV), and I = importance factor (1.0 for general structures). In the elastic analysis of structure to calculate the design forces, the size of column is considered as 500 mm 500 mm whereas beam is considered to be of dimension 300 mm 600 mm. The thickness of the shear wall is taken as 200 mm and 230 mm respectively for structure-1 and 2. Thickness of slab is considered to be 150 mm in both the cases. Factored live load of 3 kN/m2 uniformly distributed on each floor is considered in the design. Details of both structures are given in Table 3. The shear walls of the structures-1 & 2 are designated as SW-1 and SW-2 respectively. Finite element models of the structures are shown in Fig. 9 (a) and (b). Design and detailing (of shear wall) is carried out following (Indian Standards, 2000, 1993) using limit state method. Design base shear is calculated to be 560 kN and 4035 kN and design
Table 3 Details of the structures considered for case study. Structure
Response reduction factor
Design PGA
Material properties
1
5.0
2
1.0
0.024 g (DBE) 0.24 g (MCE)
Fck = 25 Mpa, Fy = 415 Mpa Fck = 30 Mpa, Fy = 415 Mpa
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(a) Structure-1
(b) Structure-2 Fig. 9. FE model of structures for case study.
(a) SW -1
(b) SW -2 Fig. 10. Reinforcement details of shear walls.
walls oriented in the direction of the motion. Contribution of columns to resist base shear is less than 2% and shear walls in other direction contributes to around 2%. As majority of the lateral load due to earthquake is taken by shear walls only, pushover analysis is performed on the shear walls instead of whole building to evaluate the lateral load deformation behavior of the structure.
moment is 9560 kN-m and 40,200 kN-m respectively for SW-1 and SW-2. Both the shear walls are provided with boundary elements having higher reinforcement at the ends. Detailing of the shear walls is done in such a way that the amount of reinforcement provided is just equal to the required reinforcement. Reinforcement detailing of walls is shown in Fig. 10. 4.2. Generation of FRS for beyond design basis seismic demand
4.2.1. Pushover analysis of shear walls FE model of the shear wall is developed in FE software Abaqus using shell element (6 DOF per node) with smeared layer of reinforcement (type S4R). Mesh size of for finite element modeling were arrived after performing a sensitivity study. Average size of shell elements representing shear walls is around 0.3 m. Nonlinear behavior of concrete is modeled using Concrete Damaged Plasticity (CDP) material model following the same approach mentioned in Section 3.1. Reinforcing steel is modeled using elasto-plastic model.
Lateral load analysis of the structure indicates that more than 95% of the base shear due to earthquake load is shared by shear Table 4 Dynamic Characteristics of shear walls. Mode
1
Frequency, Hz
% Mass Participation
SW-1
SW-2
SW-1
SW-2
1.71
6.85
69.0
71.6
10000
900
9000 8000
700
Base shear (KN)
Base Shear (kN)
800
600 500 400 300 200
7000 6000 5000 4000 3000 2000 1000
100
0
0 0
0.005
0.01
0.015
0.02
0.025
0.03
0
0.005
Roof Displacement, m
(a) SW -1
0.01
0.015
0.02
0.025
0.03
Roof displacement, m
(b) SW -2 Fig. 11. Capacity curve of shear walls.
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Pushover analysis of shear wall under monotonically increasing load is carried out with lateral load distribution corresponding to first mode (cantilever mode) and capacity curve of the wall is evaluated. Dynamic properties of shear walls for fundamental mode is given in Table 4. Capacity curve of the shear walls is shown in Fig. 11. Ultimate capacity of SW-2 is around 10 time higher than SW-1. 4.2.2. Seismic performance assessment of shear walls Seismic performance assessment of SW-1 is conducted for two cases, Design Basis Earthquake (DBE) and Maximum Considered
Sa, g
0.14 0.12
DBE
0.1
MCE
Earthquake (MCE). Response spectra for these three level of seismic motion are shown in Fig. 12. Seismic performance assessment of SW-2 is also evaluated for two level of seismic input motion. These input motions are postulated Safe Shutdown Earthquake (SSE) and Beyond Design Basis Earthquake (BDBE) for a NPP site. Response spectra for the same is shown in Fig. 13. As the structure-2 was designed for higher seismic force (design PGA of 0.24 g), input motion selected for performance assessment had PGA of 0.2 g and 0.49 g for SSE and BDBE respectively so as to produce moderate and high level of damage in the structure. Pushover curves of the walls are converted into capacity spectra and superimposed with reduced demand spectra in Acceleration Displacement Response Spectrum (ADRS) format to evaluate the performance points as shown in Fig. 14. 4.2.3. Calculation of stiffness reduction and equivalent damping Based on performance point corresponding to different levels of seismic ground motion, stiffness reduction factors and equivalent damping due to nonlinear deformation are evaluated as per the procedure mentioned previously. For estimation of equivalent viscous damping, formulation for shear wall has been applied with elastic damping of 5% as per Eq. (3). Details of the performance point, stiffness reduction factors and equivalent damping values for the shear walls are given in Table 5 below.
0.08 0.06 0.04 0.02 0 0.1
1
10
4.2.4. Generation of FRS FE model of the shear wall is modified to account for stiffness degradation and equivalent damping for corresponding level of ground motion. Modulus of elasticity of concrete for shear walls is reduced in line with the stiffness reduction factor to account for stiffness degradation. Modal time history analysis has been performed on modified 3D model of the structure including enhanced equivalent damping to generate floor time history used for calculating FRS. Input time histories are artificially generated from the ground response spectra.
100
Frequency, Hz Fig. 12. Response spectra for DBE and MCE.
1.2 1
BDBE
Sa, g
0.8
SSE
0.6 Table 5 Stiffness reduction factor and hysteretic damping for shear walls.
0.4
Parameter
0.2
SW-1
0 0.1
1
10
Roof drift, m Base shear, kN Stiffness reduction factor Equivalent damping (%)
100
Frequency, Hz Fig. 13. Response spectra for SSE and BDBE.
Capacity spectrum DBE MCE
MCE
SSE
BDBE
0.0058 282 1.0 5.00
0.013 461 0.48 5.40
0.0029 2952 0.9 5.33
0.01 4741 0.17 7.60
Capacity spectrum SSE BDBE
10 8
1
Sa (m/s2)
Sa (m/s2)
1.5
SW-2
DBE
0.5
6 4 2 0
0 0
0.005
0.01
0.015
Spectral displacement, Sd (m)
0.02
0
0.005
0.01
0.015
0.02
Spectral displacement, Sd (m)
(a) SW-1
(b) SW-2 Fig. 14. Performance assessment of shear walls.
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0.6
6 DBE MCE
0.5 0.4
BDBE
4 Sa, g
Sa, g
SSE
5
0.3
3
0.2
2
0.1
1
0
0 0.1
1 10 Frequency, Hz
100
0.1
(a) SW-1
1
10 Frequency, Hz
100
(b) SW-2 Fig. 15. FRS of the structures at top floor.
Table 6 Key parameters from FRS. Parameter
ZPA Peak Accl. (g) Frequency at peak accl. (Hz)
SW-1
SW-2
DBE
MCE
SSE
BDBE
0.07 0.37 1.71
0.17 0.55 1.34
0.70 4.86 6.33
0.90 3.88 3.19
Fig. 15 shows the FRS of the shear walls at top floor for the different levels of seismic ground motion. For SW-1, Zero Period Accelerations (ZPA) of the FRS for DBE and MCE are obtained as 0.07 g, 0.17 g respectively. Reduction in spectral amplification with increase in seismic demand is seen owing to degradation of structural stiffness and increased damping. As per conventional adopted procedures for FRS broadening (ASCE, 2000) the locations of peaks in FRS are broadened by ±15%. In the current evaluation, with increase in demand, major spectral peaks are seen to be shifted as much as 50%, indicating need for specific analysis for earthquake beyond design basis. For SW-2, stiffness degradation under SSE is considerably lower than under BDBE. ZPA for SSE and BDBE is obtained as 0.7 g and 0.9 g respectively. Peak spectral acceleration of SSE is higher than that of BDBE. Owing to predominantly high frequency content of spectra considered, the reduction in structural frequency under BDBE due to degradation is seen to result in relatively lower spectral acceleration demand compared to that for SSE. Key parameters from FRS of both the structures is given in Table 6.
has limitations arising from the method of linearization of nonlinear global response. The mismatch observed in a few local smaller peaks and peak value of top floor acceleration can perhaps be attributed to the above limitation. Based on the study following conclusions are made. 1. The proposed method is simple for design office use yet can generate FRS for high seismic demands with reasonable accuracy. 2. If there is no significant degradation of the structure under higher seismic demand (i.e. the structure has large margin over its Design Basis Earthquake), there is no change in shape of the floor response spectra (FRS) compared to the FRS based on linear analysis. Thus the amplification scales almost linearly. 3. If there is a perceptible degradation of the structure exposed to ground motion higher than its design basis (i.e. BDBE motion), the resultant floor response spectra would shift to lower frequencies and peak value of floor acceleration also reduces (depending on the frequency content of the input spectra). This is due to softening of the rigidity on account of material degradation and consequent enhancement in damping (energy absorption) of the structure. 4. Broadening of peaks of FRS using conventional approach (±15%) may not be sufficient to capture the degradation of structural stiffness/frequency under Beyond Design Basis Earthquake. 5. Further refinement of the proposed method can be made accounting for localized damage and resulting effect on FRS which is not currently captured.
5. Summary and conclusions References A simplified methodology for generation of FRS for seismic demands higher than design basis of the structure is presented in this work. The proposed methodology utilizes pushover curve of the structure to determine the nonlinear response in terms of stiffness reduction factor and hysteretic damping due to inelastic response of the structure. These parameters (stiffness reduction factor and increased hysteretic damping) are incorporated into the FE model to develop an equivalent linear model for linear time history analysis. FRS is then generated using floor time histories. The proposed methodology is validated by comparing the FRS obtained from proposed methodology for a RC structure tested on shake table, with the experimental data. After validation the methodology is applied to two sample structures designed for low and high seismic forces respectively. The proposed method
Abaqus-6.10, 2010. Abaqus 6.10 User Documentation – User Manual. Dassult Systems Simulia Crop, Providence, RI, USA. ANSYSÒ Academic Research, 2015. Release 16.2, ANSYS, Inc. ASCE, 2000. ‘‘Seismic Analysis of Safety-Related Nuclear Structures and Commentary”, ASCE 4–98. American Society of Civil Engineers, Reston, Virginia. ATC-40, 1996. ‘‘Seismic Evaluation and Retrofit of Concrete Buildings”, Report No. SSC 96–01, Applied Technology Council, California Seismic Safety Commission, California November 1996. Calvi, P.M., Sullivan, T.J., ‘‘Improved Estimation of Floor Spectra in RC Wall Buildings”. In: Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July 21–25, 2014, Anchorage, Alaska. CEN EC8, 2004. Eurocode 8 – ‘‘Design Provisions for Earthquake Resistant Structures”, EN-1998–1:2004: E, ComiteEuropeen de Normalization, Brussels, Belgium. Gregory, L., Fenves, Frank Mckenna, Silvia Mazzani, 2007. ‘‘OpenSees” Computational Simulation Services in NEESit, University of California, Berkeley, Version 1, (http://it.nees.org/support/workshops/2007/2wfcree/).
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Gülkan, P., Kazaz, I., Yakut, A., 2005. ‘‘Caveats For Nonlinear Response Assessment of Short-Period Structures”. Earthquake Engineering Research Center, Disaster Management Research Center, Department of Civil Engineering, Middle East Technical University. IAEA, 2003. ‘‘Seismic Evaluation of Existing Nuclear Power Plants”, International Atomic Energy Agency, IAEA Safety Reports Series.No.28, Vienna. IAEA CRP-NFE, Combescure, D., 2002. ‘‘Camus Benchmark: Experimental Results and Specifications to the Participants”. Rapport DM2S, SEMT/EMSI/RT/02-047/ A. Igusa, T., Der Kiureghian, A., 1985. Generation of floor response spectra including oscillator-structure interaction. Earthq. Eng. Struct. D. 13 (5), 661–676. IS-13920, 1993. Indian Standard ‘‘Ductile Detailing of Reinforced Concrete Structures Subjected to Seismic Forces”. Bureau of Indian Standards, New Delhi. IS 456, 2000. Indian Standard, ‘‘Plain and Reinforced Concrete – Code of Practice (Fourth Revision)”. Bureau of Indian Standards, New Delhi. IS-1893 (part1), 2002. Indian Standard, ‘‘Criteria for Earthquake Resistant Design of Structures, Part 1 General Provisions and Buildings (Fifth Revision)”. Bureau of Indian Standards, New Delhi. Jha, S., Bharghava, K., Bishnoi, L.R., Roshan, A.D., Basu, S., 2013. ‘‘Numerical Simulation of Shear Wall Nonlinear Behavior Subjected to Lateral Loads”. In: 22nd International Conference on Structural Mechanics in Reactor technology (SMiRT-22), San Francisco, USA, Paper # 268, Division VII. Jha, S., Bharghava, K., Bishnoi, L.R., Roshan, A.D., Basu, S., 2015. ‘‘Seismic Margin Assessment of Shear Walls with Different Aspect Ratio”. In: 23rd International Conference on Structural Mechanics in Reactor technology (SMiRT-23), Manchester, United Kingdom, Paper # 451, Division IX. Kennedy, R.P., Short, S.A., Merz, K.L., Tokarz, F.J., Idriss, I.M., Power, M.S., Sadigh, K., 1984. ‘‘Engineering Characterization of Ground Motion, Task 1: Characteristics of Free- Field Motion on Structural Response”. In: NUREG/CR-3805, Structural Mechanics Associates, Inc. and Woodward-Clyde Consultants.
Locci, J.M., et al., 1998. ‘‘CAMUS” International Benchmark –Experimental results. Synthesis of the participants, reports, Document prepared by CEA, CEN and the AFPS Benchmark Working Group, Paris, France, pp 1–77. Miranda, E., Taghavi, S., 2006. ‘‘Estimation of Seismic Demands on AccelerationSensitive Nonstructural Components in Critical Facilities”. In: Proceedings of the Seminar on Seismic Design, Performance, and Retrofit of Nonstructural Components in Critical Facilities, ATC 29–2, Newport Beach, CA, 347–360. Park, R., Priestley, M.J.N., Gill, W.D., 1982. Ductility of square confined concrete columns. Struct. Div. ASCE 108 (4), 929–950. PEER/ATC-72-1, 2010. Modeling and Acceptance Criteria for Seismic Design and Analysis of Tall Buildings. Applied Technology Council, California. Schnabel, P.B., Lysmer, J., Seed, H.B., 1972. ‘‘SHAKE: a computer program for earthquake response analysis of horizontally layered sites”, Report EERC-72-12, Earthquake Engineering Research Center. University of California, Berkeley. Seed, H.B., Idriss, I.M., 1970. Soil moduli and damping factors for dynamic response analysis”, Report EERC 70-10, Earthquake Engineering Research Center. University of California,, Berkeley. Sullivan, T.J., Calvi, P.M., Nascimbene, R., 2013. Towards Improved Floor Spectra Estimates for Seismic Design. Earthq. Struct. 4 (1). Valles, R., Reinhorn, A., Kunnath, S., Li, C., Madan, A., 1996. IDARC 2D Version 5.0: ‘‘A Program for the Inelastic Damage Analysis of Buildings”, Technical Report NCEER-96-0010, NCEER, State University of New York at Buffalo, Buffalo, NY 14261, 1996. Villaverde, R., 1997. Seismic design of secondary structures: state of the art. J. Struct. Eng. ASCE 123 (8), 1011–1019. Wahalathantri, B.L., Thambiratnam, D.P., Chan, T.H.T., Fawzia, S., 2011. ‘‘A material model for flexural crack simulation in reinforced concrete elements using ABAQUS”. First International Conference on Engineering, Designing and Developing the Built Environment for Sustainable Wellbeing, Queensland University of Technology, Brisbane, Qld, pp. 260–264.
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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Floor response spectra for beyond design basis seismic demand Somnath Jha ⇑, A.D. Roshan, L.R. Bishnoi Siting & Structural Engineering Division, Atomic Energy Regulatory Board, Mumbai 400094, India
a r t i c l e
i n f o
Article history: Received 31 May 2016 Received in revised form 23 December 2016 Accepted 4 January 2017 Available online xxxx
a b s t r a c t Vulnerability assessment of Structures Systems and Components (SSCs) of nuclear facilities for earthquake ground motion exceeding the design basis ground motion has become a key issue to ensure safety in case of highly improbable but possible extreme earthquake event beyond design basis. In order to assess seismic safety of SSCs located at different floors of the building, floor response spectra (FRS) for such beyond design basis seismic demand, taking into account possible nonlinear behavior of structure, is required. Currently used methods (e.g. IAEA Safety Series 28) do not capture softening of structure at increased demand. A performance based approach would be ideal under such situations. FRS for higher seismic demand may be developed through nonlinear time history analysis of the structure. However, this approach is complicated and cumbersome for structural configurations such as that of NPPs and requires deep insight of solution algorithm and large resources for numerical solution. In this work a simplified methodology for FRS generation for higher level of seismic demand using linear time history analysis of an equivalent linear structural model is proposed. Nonlinear behavior of structure and associated damping are estimated from Nonlinear Static Pushover Analysis. Through this approach realistic estimation of stiffness degradation of the structure and increase in damping at different levels of seismic demands can be made. Subsequently, linear time history analysis is performed on degraded structure, accounting for stiffness reduction and hysteretic damping to obtain the floor time histories which are then converted to FRS. Application of this method is shown through case studies. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Seismic safety is one of the essential design requirements of a NPP structure. Occurrence of recent seismic events (NCO earthquake of 2007 and GEJE event of 2011) that resulted in ground motions beyond design basis at many NPP sites has emphasized the need for assessment of margins to ensure safety against beyond design basis events. With increase in demand, structure may exhibit nonlinear behavior resulting in softening (stiffness reduction) and change in natural frequencies. Nonlinear behavior of the structure also results in additional damping due to hysteresis. Conventional approaches (e.g. IAEA, 2003) address this as an
Abbreviations: ADRS, Acceleration Displacement Response Spectrum; BDBE, Beyond Design Basis Earthquake; CDP, Concrete Damaged Plasticity; CSM, Capacity Spectrum Method; DBE, Design Basis Earthquake; FE, Finite element; FRS, floor response spectra; GEJE, Great East Japan Earthquake; MCE, Maximum Credible Earthquake; NPP, Nuclear Power Plant; NLSPA, Nonlinear Static Pushover Analysis; PGA, Peak Ground Acceleration; SSCs, Structures Systems & Components; SSE, Safe Shutdown Earthquake. ⇑ Corresponding author. E-mail address: [email protected] (S. Jha).
extension of existing seismic analysis methodologies albeit with use of increased damping and response reduction factors due to ductility. Performance assurance of floor mounted equipment and components is important for NPPs even under Beyond Design Basis Earthquake events. Seismic performance of these systems can only be assured if seismic demands on floors is appropriately quantified. Seismic demands on floors is generally represented through Floor response spectra (FRS). FRS for Design Basis Earthquake is developed by linear time history analysis of the structure. Under BDBE, when structure is expected to exhibit inelastic behavior, ideally non-linear time history analysis would be required. Though non-linear time history analysis is a more rational method, its complexity of response tracking, uncertainties associated with material constitutive laws and their impact on solution accuracy are the deterrents for design office use of nonlinear time history analysis. Generally, only specific purpose software like OpenSees (Gregory et al., 2007), IDARC (Valles et al., 1996) can simulate the complex behavior of concrete under cyclic loading. In addition, use of any numerical system to Nuclear Power Plants calls for stringent software related quality assurance (QA) requirements.
http://dx.doi.org/10.1016/j.nucengdes.2017.01.006 0029-5493/Ó 2017 Elsevier B.V. All rights reserved.
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Symbols be b bh b0 ED ESo fs f
effective damping viscous damping of shear wall structure pinched hysteresis damping hysteresis damping energy dissipated by damping maximum strain energy secant frequency elastic frequency
Different simplified approaches, as via media, are proposed in the literature for estimation of seismic demands on floors (Igusa and Der Kiureghian, 1985; Villaverde, 1997; Miranda and Taghavi, 2006; CEN EC8, 1998) among others). Limitations of these methods have been extensively reviewed by Sullivan et al. (2013). It is noted that, most approaches in the literature do not consider the impact of different levels of elastic damping and non-linear behavior of the primary structure on floor spectra. These approaches are either simple but not reliable or reliable but require relatively advanced analysis capabilities (Calvi and Sullivan, 2014). Calvi and Sullivan (2014) proposed a simplified method for generation of FRS and validated it by comparing predicted spectra with those obtained from time-history analyses of a case study building. The method proposed in Calvi and Sullivan (2014) could be applied to multi-story structures only in linear range, but it is possible to adjust elastic damping values over a wide range. Development of a simplified approach, which is capable of addressing the non-linear behavior of the primary structure and increased damping that influences the shape and intensity of floor spectra, is quite challenging. The methodology proposed in this work for generation of FRS utilizes state of the art seismic performance assessment procedures viz. Nonlinear Static Pushover Analysis (NLSPA) for accounting non-linear behavior of the structure. Simplicity of pushover analysis makes it more appealing for design office use to calculate degraded stiffness of the structure at different levels of seismic demand exceeding design basis. Hysteretic damping due to nonlinear behavior of the structure could also be evaluated from pushover curve using empirical approaches given in various international guidelines (ATC-40, 1996; Kennedy et al., 1984). The main postulation of the proposed method is that linear time history analysis on equivalent linear structure, after accounting for stiffness degradation and enhanced hysteretic damping can yield floor response spectra for seismic demand, corresponding to Beyond Design Basis Earthquake (BDBE). Industrial standard software platform such as ANSYS (2015) and Abaqus (2010) are being used in nuclear applications after satisfying software quality assurance requirements. The implementation of proposed methodology is shown through the use of Abaqus software. 2. Methodology for generation of FRS using pushover analysis NPP structures may exhibit nonlinear behavior if subjected to ground motion higher than the design basis. FRS required for seismic qualification of floor mounted equipment and components for beyond design basis ground motion should be developed taking into account possible degradation of the structure and expected level of damping. Static pushover analysis is a tool by which stiffness degradation of the structure and expected damping at different levels of seismic demand can be evaluated.
fe KS KI Z I R SRF
effective frequency secant stiffness initial stiffness zone factor (0.24 for Seismic Zone-IV of IS 1893 (2002)) importance factor (1.0 for conventional structure) response reduction factor (5.0 for Shear walls) stiffness reduction factor
In this work, pushover analysis is proposed to be used to determine stiffness degradation of structure and modified damping at different levels of seismic demand. This is done by seismic performance assessment of the structure using Capacity Spectrum Method of ATC-40 (1996). Finite element (FE) model of structure is then modified to take into account the stiffness degradation and modified damping expected at any specified level of seismic motion beyond design basis. Modified FE model of the structure may be named as equivalent linear model. Linear time history analysis of the degraded structure (reduced stiffness) is performed for the modified damping to generate FRS. 2.1. Seismic performance assessment In Capacity Spectrum Method, capacity of structure is compared with seismic demand in Acceleration Displacement Response Spectrum (ADRS) format to assess the seismic performance of structure for a given seismic motion. The procedure seeks to find out a point on the capacity spectrum that also lies on the appropriate demand spectrum, scaled down for nonlinear effects. This point is called performance point. Hysteretic damping and stiffness degradation for the ground motion under consideration is estimated in correspondence of the evaluated performance point. 2.2. Estimation of hysteretic damping Damping that occurs when the earthquake ground motion derives the structure into inelastic range can be viewed as the combination of viscous damping that is inherent in the structure and the hysteretic damping. Hysteretic damping is dependent on area of hysteretic loops formed when the earthquake force (base shear) is plotted against the structure displacement. Hysteretic damping can be represented by equivalent viscous damping using equations available in the literature (ATC-40, 1996; Kennedy et al., 1984; Gülkan et al., 2005). An approximate effective (hysteretic) damping is calculated based on the capacity curve, the estimated displacement demand and resulting hysteresis loop. Probable imperfections in real building hysteresis loops, including degradation and duration effects, are accounted by reducing theoretically calculated equivalent viscous damping values. Equivalent viscous damping of frame structures (i.e. with beam and columns), can be estimated using bilinear hysteretic model given in ATC-40 (1996) and reproduced as Eq. (1). Based on studies of nonlinear response of short period structures, Gülkan et al. (2005) reported that equation given in ATC-40 significantly underestimates the displacement demand of short period structures such as shear walls due to overestimation of viscous damping. High damping values are not expected for shear wall structures due to considerably lower ultimate displacement demand than frame structures and existence of pinching in the hysteresis loop. For experiments conducted on shake table under significant damage
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S. Jha et al. / Nuclear Engineering and Design xxx (2017) xxx–xxx
conditions, the maximum value of damping in RC wall-frame system ranges from 6.9% to 7.5% (PEER/ATC-72-1, 2010), whereas damping predicted by equation given in ATC-40 results in values as high as 20% in some cases. Recognizing this problem, some researchers have proposed equations for estimation of equivalent viscous damping of shear wall structures. Jha et al. (2015) performed a study on prediction of equivalent viscous damping of shear walls by different formulations available in literature. Outcome of the study shows that formulation proposed by Kennedy et al. (1984) (reproduced as Eq. (3)) accounts for structure specific characteristics of shear walls. Thus this formulation was selected for current study. More details on comparison of damping from different formulations and validation of equation used in the current study are available in Jha et al. (2015). 2.2.1. Damping equation as per ATC-40
be ¼ ðb0 þ 0:05Þ
ð1Þ
where
b0 ¼
1 ED 2ðl 1Þð1 aÞ ¼k 4p ESo plð1 þ al aÞ
ð2Þ
wherel ¼ Displacement ductility ratio, a = post elastic stiffness ratio, and k = factor to account structure type. 2.2.2. Damping equation as per Kennedy et al. (1984)
be ¼
fs f fe f
!2 ðb þ bh Þ
ð3Þ
wherebe ¼ effective damping, b ¼ viscous damping of shear wall structure, bh ¼ pinched hysteresis damping, f s ¼ secant frequency, f ¼ elastic frequency, f e ¼ effective frequency.
f bh ¼ 0:11 1 s f fs ¼ f
ð4Þ
rffiffiffiffiffi Ks K
ð5Þ
where K and KS are initial and secant stiffness obtained from pushover curve (See Fig. 1).
fe f ¼ ð1 AÞ þ A s f f
ð6Þ
where A ¼ C F ð1 ffs Þ 6 0:85; &C F ¼ 2:3
Tangent Sffness, K T
Details of formulation to estimate these parameters can be found in ATC-40 and Kennedy et al. (1984). 2.3. Estimation of stiffness degradation Capacity curve obtained using pushover analysis represents degradation of structural stiffness under lateral (seismic) load. Performance point obtained using well established procedures such as Capacity Spectrum Method indicates the maximum level of degradation under seismic demand imposed on the structure. Hence, actual stiffness of the structure at performance point is represented using tangent drawn on the pushover curve at performance point. This is pictorially shown in Fig. 1. It is worth noting that the structure has reached at this point by progressive damage under seismic loading. To derive stiffness degradation from capacity curve to develop equivalent linear model, this progressive degradation of the structure needs to be considered. Equivalent linear methods are most commonly used in Soil Dynamics and Geotechnical Earthquake Engineering for ‘‘site response analysis” in which the free field acceleration of a site with stratified soil deposit is computed using earthquake induced motion at the bedrock. Soil layers subjected to ground motion undergo nonlinear deformations, which needs to be account for their nonlinear behavior. Although ‘‘site response analysis” can be performed using rigorous nonlinear step-by-step time history analysis, this is not the approach followed in practical applications, owing to various uncertainties and computational complexities. The limitations are similar to performing nonlinear time history analysis on reinforced concrete structures. As a general and widely accepted practice, ‘‘equivalent linear method” originally proposed by Seed and Idriss (1970) and implemented in the well-known programs such as SHAKE (Schnabel et al., 1972) is used to calculate the seismic response of the soil deposits. Nonlinear behavior of the soil is approximated by using equivalent shear modulus and damping ratios that are function of effective shear strains. The effective strain is defined by reducing the maximum strain response. A reduction factor of 0.65 is used over the maximum strain value to account for progressive degradation. Feasibility of applying concept of equivalent linear method used in Geotechnical Earthquake Engineering for the present work is assessed. In order to account for progressive degradation of the structure till the performance point, a factor of 1/0.65, is applied on to the degraded stiffness. In other words, to evaluate stiffness of equivalent linear model, tangent stiffness at performance point is increased by a ratio, 1/0.65 (see Fig. 1), to account for the fact that the structural stiffness has degraded to tangent stiffness from initial stiffness, progressively. Thus stiffness reduction factor (SRF) can be obtained by comparing the equivalent stiffness with initial stiffness of the structure. Equation to evaluate SRF is given below.
SRF ¼
Performance point Equivalent Sffness, K T/0.65
Inial Sffness, K I
Fig. 1. Representation of equivalent stiffness.
3
KT 0:65 K I
ð7Þ
This reduction factor is applied to stiffness of all the structural components that contributed to the nonlinear behavior under the specified seismic demand such that the overall structural stiffness is representative of the equivalent softened structure at performance point. This methodology has limitations as the stiffness reduction is not implimented locally to individual members and does not capture progressive softening. However, this approximation would still be reasonable in view of the fact that any local reduction of stiffness would result in redistribution of demand among individual members and thus degrading the global behavior.
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3. Validation of the proposed methodology The proposed methodology is validated for a reinforced concrete structure by comparing experimental results of shake table experiments carried out in the Saclay Nuclear Center in France in 1997 (Locci and et al., 1998). Specimen CAMUS was a 1/3-scale model of a representative 5-story reinforced concrete building detailed according to French practice (IAEA CRP-NFE, 2002). CAMUS specimens were designed considering it as a typical example for a stiff structure commonly used in NPPs. It had two parallel shear walls linked by 6 square floors (1.7 m 1.7 m) (including the floor connected to the footing). The specimen had a total mass of 36 tons with additional masses attached to it. A heavily reinforced concrete footing allowed anchorage to the shaking table. The total height of the model was 5.10 m. Shear walls had a width of 1.7 m and thickness of 6 cm. The stiffness and the strength of the structure in direction perpendicular to shear walls are increased by adding some lateral triangular bracing. The specimen had a measured fundamental frequency of 7.24 Hz and the first vertical eigen mode had a natural frequency of 20 Hz. The dimensions of the specimen are shown in Fig. 2. Other details including seismic weights on floors, reinforcement detailing and material properties etc. can be found in IAEA CRP-NFE (2002). The structure was tested on shake table with mono-axial seismic excitations (parallel to shear wall) with increasing level of input motions. Result of first applied input motion with PGA of 0.25 g is considered for validation in present study. Time history and response spectra of the input motion considered in the experiment are shown in Fig. 3. 3.1. Finite element modeling A 3-D model of the structure is developed in finite element software Abaqus. The actual material properties and boundary conditions in the experiment were implemented in the model to reflect the required aspects of the test specimen. Finite element
model of the structure is shown in Fig. 4. Shear walls, footing and slabs are modeled using 4-noded shell elements (S4R). Triangular bracing provided to prevent non-symmetric failure is modeled using 2-noded beam elements (B31). Average size of finite elements for shear walls was around 0.1 m. This number was arrived at after performing mesh sensitivity studies. Nonlinear behavior of concrete is modeled using Concrete Damaged Plasticity (CDP) material model of Abaqus. This material model has been calibrated and validated for prediction of inelastic behavior of shear wall under lateral loads by comparing experimental data obtained from test conducted on shear walls with varying aspect ratio (Jha et al., 2013). Park et al. (1982) model is used for stress strain curve of concrete under compression and the model proposed by Wahalathantri et al. (2011) is used for tension stiffening. Elasto-plastic model with actual yield and ultimate strength (from experiment) is used for reinforcing steel. 3.2. Pushover analysis of the structure Pushover analysis of the structure is performed using load proportional to fundamental mode. Natural frequency of the structure is calculated to be 7.245 Hz which compares closely with the experimentally observed frequency of 7.24 Hz. Frequency of first vertical mode is calculated to be 20.04 Hz as against measured frequency of 20 Hz. Pushover curve of the structure up to top deflection of 10 mm is shown in Fig. 5. 3.3. Seismic performance assessment Performance point of the structure is evaluated using Capacity Spectrum Method. Hysteretic damping for performance assessment is calculated using Eq. (3) (see Section 2.2.2). Comparison of capacity and demand in ADRS format is shown in Fig. 6. Maximum roof deflection of 7 mm at a base shear of 140 kN was obtained in the experiment. Seismic performance assessment in present study predicted a maximum roof drift of around 6.4 mm
Fig. 2. Camus Specimen (IAEA CRP-NFE, 2002).
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1.2
0.3
1
Sa, g
Acceleraon, g
0.2 0.1 0 0
10
20
30
-0.1
40
0.8 0.6 0.4
Time, s 0.2
-0.2
0
-0.3 0.1
1
10
100
Frequency, Hz Fig. 3. Time history and response spectra of input motion.
Spectral Acceleraon, Sa
10 7.5 5 2.5 0 0
0.0025 0.005 0.0075 Spectral Displacement, Sd
0.01
Fig. 6. Comparison of capacity and demand in ADRS format for CAMUS specimen. Fig. 4. Finite element model of CAMUS specimen. Table 1 Calculation of hysteretic damping for CAMUS specimen.
Base Shear, kN
200 160
Ks K
fs f
A
fe f
be (%)
0.6
0.77
0.53
0.88
5.78
120 80 Table 2 Calculation of stiffness reduction factor (SRF) for CAMUS specimen.
40 0 0
2
4 6 Deflecon, mm
8
10
Fig. 5. Pushover curve of CAMUS specimen.
at a base shear of 155 kN. Thus, performance point of the structure could be predicted with reasonable accuracy. 3.4. Estimation of hysteretic damping and stiffness reduction factor 3.4.1. Hysteretic damping Calculation for hysteretic damping using Eq. (3) at performance point is given in Table 1. 3.4.2. Stiffness reduction factor Calculation for stiffness reduction factor at performance point is given in Table 2.
KT (kN/m)
KI (kN/m)
SRF
1.19E + 04
4.09E + 04
0.446
3.5. FRS using proposed methodology As per the proposed methodology, equivalent linear model of the structure is developed using stiffness reduction factor and hysteretic damping. A linear modal time history analysis is performed for the input motion of the shake table shown in Fig. 3 and corresponding FRS is generated for different floors of the structure. The result is then compared with the FRS obtained from floor time histories recorded during the experiment. As shown in Figs. 7 and 8, FRS obtained using the proposed methodology based on equivalent linear model for different floors are in good agreement with the experimental FRS. The minor differences in some local peaks (Fig. 7a and b) are attributed to the linearization of the non-linear response. The dif-
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1.2
2 Experimental
Equivalent linear analysis Experimental
1
1.6
Equivalent Linear Analysis
0.8
Sa, g
Sa, g
1.2 0.6
0.8
0.4 0.4
0.2 0 0.1
0 1
10
100
Frequency, Hz
0.1
1
(a) 2nd Floor
10
Frequency, Hz
100
(b) 3rd Floor
Fig. 7. Comparison of FRS from experiment and analysis for 2nd and 3rd floor.
6 Experiemental 5 Equivalent linear analysis
Sa, g
4 3 2 1 0 0.1
1
Frequency, Hz
10
100
Fig. 8. Comparison of FRS from experiment and analysis for 6th (top) floor.
ference in the peak value for the top floor (Fig. 8) needs further investigation. 4. Case studies for generation of FRS Owing to large loads and / or shielding considerations, many of the NPP structures are characterized by shear wall type systems. To enable assessment of systems with such configurations, the methodology proposed in this work for generation of FRS under conditions beyond design basis is applied to two sample RCC structures with shear walls. Impact of change in aspect ratio of the wall is also captured in the assessment. 4.1. General details of sample structures Structures considered for the case study consists of two full scale 5 and 3-story buildings with shear walls as main load resisting members. Both the structures have plan dimensions of 24 m 24 m, grid size of 6 m 6 m and floor height of 4 m. Four shear walls are located on the periphery having width of 4 m and 6 m respectively for structure-1 & 2. Indian standard code (2002) has been used for design of these structures. This standard specifies two levels of earthquakes, one is called as Design Basis Earthquake (DBE) and another as maximum considered earthquake. Four seismic zones are defined in this standard starting from
Zone-II (lowest seismicity) to Zone-V (highest seismicity). First structure is designed for Design Basis Earthquake and assumed to be located in a high seismic zone (Zone-IV). Response reduction factor, R, is taken as 5.0 for structure-1. The second structure, three storied RC shear wall-frame building, is considered to be located in same seismic zone but designed for maximum considered earthquake (MCE), which has PGA twice of DBE. Response reduction factor for the second structure is taken as unity in line with principle followed for NPP structures. These structures are designed using two different approaches (different response reduction factor) to demonstrate versatility of the proposed method for different level of damage. Structure-1, designed for lower seismic loads, is expected to suffer more damage when subjected to same level of seismic motion. PGA for Design Basis Earthquake (DBE) is evaluated using following equation given in Indian standard code (2002):
PGA ¼
Z I 2 R
ð8Þ
where Z = seismic zone factor (0.24 for Zone-IV), and I = importance factor (1.0 for general structures). In the elastic analysis of structure to calculate the design forces, the size of column is considered as 500 mm 500 mm whereas beam is considered to be of dimension 300 mm 600 mm. The thickness of the shear wall is taken as 200 mm and 230 mm respectively for structure-1 and 2. Thickness of slab is considered to be 150 mm in both the cases. Factored live load of 3 kN/m2 uniformly distributed on each floor is considered in the design. Details of both structures are given in Table 3. The shear walls of the structures-1 & 2 are designated as SW-1 and SW-2 respectively. Finite element models of the structures are shown in Fig. 9 (a) and (b). Design and detailing (of shear wall) is carried out following (Indian Standards, 2000, 1993) using limit state method. Design base shear is calculated to be 560 kN and 4035 kN and design
Table 3 Details of the structures considered for case study. Structure
Response reduction factor
Design PGA
Material properties
1
5.0
2
1.0
0.024 g (DBE) 0.24 g (MCE)
Fck = 25 Mpa, Fy = 415 Mpa Fck = 30 Mpa, Fy = 415 Mpa
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(a) Structure-1
(b) Structure-2 Fig. 9. FE model of structures for case study.
(a) SW -1
(b) SW -2 Fig. 10. Reinforcement details of shear walls.
walls oriented in the direction of the motion. Contribution of columns to resist base shear is less than 2% and shear walls in other direction contributes to around 2%. As majority of the lateral load due to earthquake is taken by shear walls only, pushover analysis is performed on the shear walls instead of whole building to evaluate the lateral load deformation behavior of the structure.
moment is 9560 kN-m and 40,200 kN-m respectively for SW-1 and SW-2. Both the shear walls are provided with boundary elements having higher reinforcement at the ends. Detailing of the shear walls is done in such a way that the amount of reinforcement provided is just equal to the required reinforcement. Reinforcement detailing of walls is shown in Fig. 10. 4.2. Generation of FRS for beyond design basis seismic demand
4.2.1. Pushover analysis of shear walls FE model of the shear wall is developed in FE software Abaqus using shell element (6 DOF per node) with smeared layer of reinforcement (type S4R). Mesh size of for finite element modeling were arrived after performing a sensitivity study. Average size of shell elements representing shear walls is around 0.3 m. Nonlinear behavior of concrete is modeled using Concrete Damaged Plasticity (CDP) material model following the same approach mentioned in Section 3.1. Reinforcing steel is modeled using elasto-plastic model.
Lateral load analysis of the structure indicates that more than 95% of the base shear due to earthquake load is shared by shear Table 4 Dynamic Characteristics of shear walls. Mode
1
Frequency, Hz
% Mass Participation
SW-1
SW-2
SW-1
SW-2
1.71
6.85
69.0
71.6
10000
900
9000 8000
700
Base shear (KN)
Base Shear (kN)
800
600 500 400 300 200
7000 6000 5000 4000 3000 2000 1000
100
0
0 0
0.005
0.01
0.015
0.02
0.025
0.03
0
0.005
Roof Displacement, m
(a) SW -1
0.01
0.015
0.02
0.025
0.03
Roof displacement, m
(b) SW -2 Fig. 11. Capacity curve of shear walls.
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Pushover analysis of shear wall under monotonically increasing load is carried out with lateral load distribution corresponding to first mode (cantilever mode) and capacity curve of the wall is evaluated. Dynamic properties of shear walls for fundamental mode is given in Table 4. Capacity curve of the shear walls is shown in Fig. 11. Ultimate capacity of SW-2 is around 10 time higher than SW-1. 4.2.2. Seismic performance assessment of shear walls Seismic performance assessment of SW-1 is conducted for two cases, Design Basis Earthquake (DBE) and Maximum Considered
Sa, g
0.14 0.12
DBE
0.1
MCE
Earthquake (MCE). Response spectra for these three level of seismic motion are shown in Fig. 12. Seismic performance assessment of SW-2 is also evaluated for two level of seismic input motion. These input motions are postulated Safe Shutdown Earthquake (SSE) and Beyond Design Basis Earthquake (BDBE) for a NPP site. Response spectra for the same is shown in Fig. 13. As the structure-2 was designed for higher seismic force (design PGA of 0.24 g), input motion selected for performance assessment had PGA of 0.2 g and 0.49 g for SSE and BDBE respectively so as to produce moderate and high level of damage in the structure. Pushover curves of the walls are converted into capacity spectra and superimposed with reduced demand spectra in Acceleration Displacement Response Spectrum (ADRS) format to evaluate the performance points as shown in Fig. 14. 4.2.3. Calculation of stiffness reduction and equivalent damping Based on performance point corresponding to different levels of seismic ground motion, stiffness reduction factors and equivalent damping due to nonlinear deformation are evaluated as per the procedure mentioned previously. For estimation of equivalent viscous damping, formulation for shear wall has been applied with elastic damping of 5% as per Eq. (3). Details of the performance point, stiffness reduction factors and equivalent damping values for the shear walls are given in Table 5 below.
0.08 0.06 0.04 0.02 0 0.1
1
10
4.2.4. Generation of FRS FE model of the shear wall is modified to account for stiffness degradation and equivalent damping for corresponding level of ground motion. Modulus of elasticity of concrete for shear walls is reduced in line with the stiffness reduction factor to account for stiffness degradation. Modal time history analysis has been performed on modified 3D model of the structure including enhanced equivalent damping to generate floor time history used for calculating FRS. Input time histories are artificially generated from the ground response spectra.
100
Frequency, Hz Fig. 12. Response spectra for DBE and MCE.
1.2 1
BDBE
Sa, g
0.8
SSE
0.6 Table 5 Stiffness reduction factor and hysteretic damping for shear walls.
0.4
Parameter
0.2
SW-1
0 0.1
1
10
Roof drift, m Base shear, kN Stiffness reduction factor Equivalent damping (%)
100
Frequency, Hz Fig. 13. Response spectra for SSE and BDBE.
Capacity spectrum DBE MCE
MCE
SSE
BDBE
0.0058 282 1.0 5.00
0.013 461 0.48 5.40
0.0029 2952 0.9 5.33
0.01 4741 0.17 7.60
Capacity spectrum SSE BDBE
10 8
1
Sa (m/s2)
Sa (m/s2)
1.5
SW-2
DBE
0.5
6 4 2 0
0 0
0.005
0.01
0.015
Spectral displacement, Sd (m)
0.02
0
0.005
0.01
0.015
0.02
Spectral displacement, Sd (m)
(a) SW-1
(b) SW-2 Fig. 14. Performance assessment of shear walls.
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0.6
6 DBE MCE
0.5 0.4
BDBE
4 Sa, g
Sa, g
SSE
5
0.3
3
0.2
2
0.1
1
0
0 0.1
1 10 Frequency, Hz
100
0.1
(a) SW-1
1
10 Frequency, Hz
100
(b) SW-2 Fig. 15. FRS of the structures at top floor.
Table 6 Key parameters from FRS. Parameter
ZPA Peak Accl. (g) Frequency at peak accl. (Hz)
SW-1
SW-2
DBE
MCE
SSE
BDBE
0.07 0.37 1.71
0.17 0.55 1.34
0.70 4.86 6.33
0.90 3.88 3.19
Fig. 15 shows the FRS of the shear walls at top floor for the different levels of seismic ground motion. For SW-1, Zero Period Accelerations (ZPA) of the FRS for DBE and MCE are obtained as 0.07 g, 0.17 g respectively. Reduction in spectral amplification with increase in seismic demand is seen owing to degradation of structural stiffness and increased damping. As per conventional adopted procedures for FRS broadening (ASCE, 2000) the locations of peaks in FRS are broadened by ±15%. In the current evaluation, with increase in demand, major spectral peaks are seen to be shifted as much as 50%, indicating need for specific analysis for earthquake beyond design basis. For SW-2, stiffness degradation under SSE is considerably lower than under BDBE. ZPA for SSE and BDBE is obtained as 0.7 g and 0.9 g respectively. Peak spectral acceleration of SSE is higher than that of BDBE. Owing to predominantly high frequency content of spectra considered, the reduction in structural frequency under BDBE due to degradation is seen to result in relatively lower spectral acceleration demand compared to that for SSE. Key parameters from FRS of both the structures is given in Table 6.
has limitations arising from the method of linearization of nonlinear global response. The mismatch observed in a few local smaller peaks and peak value of top floor acceleration can perhaps be attributed to the above limitation. Based on the study following conclusions are made. 1. The proposed method is simple for design office use yet can generate FRS for high seismic demands with reasonable accuracy. 2. If there is no significant degradation of the structure under higher seismic demand (i.e. the structure has large margin over its Design Basis Earthquake), there is no change in shape of the floor response spectra (FRS) compared to the FRS based on linear analysis. Thus the amplification scales almost linearly. 3. If there is a perceptible degradation of the structure exposed to ground motion higher than its design basis (i.e. BDBE motion), the resultant floor response spectra would shift to lower frequencies and peak value of floor acceleration also reduces (depending on the frequency content of the input spectra). This is due to softening of the rigidity on account of material degradation and consequent enhancement in damping (energy absorption) of the structure. 4. Broadening of peaks of FRS using conventional approach (±15%) may not be sufficient to capture the degradation of structural stiffness/frequency under Beyond Design Basis Earthquake. 5. Further refinement of the proposed method can be made accounting for localized damage and resulting effect on FRS which is not currently captured.
5. Summary and conclusions References A simplified methodology for generation of FRS for seismic demands higher than design basis of the structure is presented in this work. The proposed methodology utilizes pushover curve of the structure to determine the nonlinear response in terms of stiffness reduction factor and hysteretic damping due to inelastic response of the structure. These parameters (stiffness reduction factor and increased hysteretic damping) are incorporated into the FE model to develop an equivalent linear model for linear time history analysis. FRS is then generated using floor time histories. The proposed methodology is validated by comparing the FRS obtained from proposed methodology for a RC structure tested on shake table, with the experimental data. After validation the methodology is applied to two sample structures designed for low and high seismic forces respectively. The proposed method
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Please cite this article in press as: Jha, S., et al. Floor response spectra for beyond design basis seismic demand. Nucl. Eng. Des. (2017), http://dx.doi.org/ 10.1016/j.nucengdes.2017.01.006