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Evaluation of seismic fragility of structures—a case study
Nuclear Engineering and Design 212 (2002) 253– 272 www.elsevier.com/locate/nucengdes
Evaluation of seismic fragility of structures —a case study Kapilesh Bhargava a,*, A.K. Ghosh b, M.K. Agrawal b, R. Patnaik a, S. Ramanujam a, H.S. Kushwaha b a
Architecture and Ci6il Engineering Di6ision, Bhabha Atomic Research Centre, Mumbai 400 085, India b Reactor Safety Di6ision, Bhabha Atomic Research Centre, Mumbai, India
Abstract The present paper attempts to evaluate the seismic fragility for a typical elevated water-retaining structure. The structure is analysed for two cases: (i) empty tank; and (ii) tank filled with water. The various parameters that could affect the seismic structural response include material strength of concrete and reinforcing steel, effective prestress available in the tank, ductility ratio and structural damping available within the structure, normalised ground motion response spectral shape, foundation and surrounding soil parameters and the total height of water available in the tank. Based on this case study, the seismic fragility of the structure is developed. The results are presented as families of conditional probability curves plotted against peak ground acceleration (PGA) at two critical locations. The procedure adopted, incorporates the various randomness and uncertainty associated with the parameters under consideration. © 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The safety of a nuclear power plant (NPP) depends upon a number of factors — intrinsic and external to the plant. Seismic ground motion is an important consideration in evaluating the safety of the plant or, alternatively, the risk associated with it. The various randomness and uncertainties associated with the occurrence of earthquakes and the consequences of their effects on the NPP components and structures call for a probabilistic risk assessment (PRA). Probabilistic seismic risk analysis (PSRA) have been or being carried out * Corresponding author. Tel.: + 91-22-5505050x22494; fax: + 91-22-5505151/5519613. E-mail address: kapil – [email protected] (K. Bhargava).
for various NPPs all over the world (Kennedy et al., 1980). Many of the existing nuclear facilities were designed and constructed during the time when seismic design procedure was evolving. Mostly, these plants were designed using the seismic coefficient method. In recent years, an increasing knowledge in the geoscience has led to a better understanding of seismic ground motion. It is also known that older plants were designed to seismic criteria that were less rigorous than those used for recent plants. In addition to this, utilities and regulatory agencies are making continuous efforts to modernise and update the structural safety due to increase in evidence of seismic hazard not envisaged earlier. The primary objective of the seismic review of the older plants is to make an overall safety assessment of the seismic capabili-
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ties of these plants, and, if necessary, to upgrade them to ensure that these plants can be safely shut down in the event of a design basis earthquake. For demonstrating the overall safety of the plant, both regulatory agencies and plant designers together have to develop a review basis ground motion (RBGM) or seismic margin earthquake (SME). A general definition of seismic margin is expressed in terms the earthquake peak ground acceleration (PGA) that compromises plant safety, specifically leading to melting of reactor core. Seismic margin assessment (SMA) involves the review of the capacity of the plant vis-a`-vis the target earthquake input (RBGM or SME) selected for review. The objectives are then to show whether the plant can withstand the effects of this SME with high confidence and to identify seismic vulnerabilities. This can be fulfilled using the results and insights obtained from PSRA, the data on actual performance of structures and equipment in actual earthquakes and tests. The key elements of PSRA comprise the evaluation of the following parameters considering the variations due to their randomness and uncertainties. (i) seismic hazard analysis at the site. (ii) Response of plant systems and structures. (iii) Seismic fragility evaluation. (iv) Systems or accident sequence analysis. (v) Evaluation of core damage frequency. It may be mentioned here that the items (ii) and (iii) are intrinsically related to each other. The seismic PRA would provide answers regarding the seismic capacities of the systems, structures and components of the plant. The large uncertainties in the seismic hazard curves make decisions regarding seismic capacity difficult whereas, the SMA review is mostly focussed on the few structures and components in the plant whose failure would lead to severe core damage. The output of SMA review is an estimation of the plant seismic capacity whereas, the seismic PRA provides estimation of seismic risks of core damage and adverse public effects. The present paper is concerned with the response of an existing elevated water-retaining structure to seismic excitation and the evaluation of seismic fragilities at two critical locations for bending mode of failure. The methodology adopted in the present
case study would be useful in the seismic reassessment of existing structures.
2. Seismic fragility and fragility model The seismic fragility of a structure may be defined as the conditional probability of its failure for a given value of the seismic input parameter e.g. PGA. The failure could be defined in terms of a limiting value of stress and/or displacement. Estimation of this PGA value, also known as ground acceleration capacity of the structure, is generally carried out by using the information available on the structural design bases, structural configurations, material properties and seismic structural response evaluated at the design stage of the structure. It may be mentioned that, there are many sources of randomness and uncertainty, that may affect the accurate estimation of the structural capacity, and this, may, in turn, affect the ground acceleration capacity for the structure significantly. Therefore, the seismic fragility is generally described by means of a family of fragility curves along with a probability value assigned to each curve to reflect the confidence level associated with the estimation of fragility. The entire fragility family for a structure corresponding to a particular mode of failure can be expressed in terms of the median ground acceleration capacity (A( ) and two random variables (mR and mU) as follows (Kennedy et al., 1980). A= A( mRmU
(1)
Here, mR and mU are the random variables with unit median values and represent the inherent randomness about the median value and the uncertainty in the median value respectively. Both mR and mU are assumed to be log-normally distributed with logarithmic standard deviations (S.D.) iR and iU respectively. The uncertainty associated with the fragility evaluation is generally expressed in terms of a range of failure probabilities for a given value of ground acceleration. The probability (P) that the conditional failure frequency (Pf) exceeds a certain specified value, P %f for a ground acceleration ‘a’ is given by (Kennedy et al., 1980).
P
n
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PfP%f ln{a/A( exp[iR = iU a
−1
n
(P%f)]}
(2)
Alternatively, the conditional failure frequency P %f at a non-exceedence probability (Q) can be expressed as (Kennedy and Ravindra, 1984): P%f =
ln(a/A( ) + iU − 1(Q) iR
Q= 1 − P
n
(3) (4)
here, (x), is the standard Gaussian cumulative distribution function; − 1 (P), the inverse of the standard Gaussian cumulative distribution function. For the purpose of displaying the fragility curves, the non-exceedence probability level ‘Q’ is utilised. In some applications, the composite variability iC is used which is defined by the following (Kennedy et al., 1980): iC = i 2R +i 2U
(5)
It may be mentioned here that, the use of iC and A( provides a single best estimate fragility curve which does not explicitly separate out uncertainty from underlying randomness. In the estimation of fragility parameters, it is convenient to evolve an intermediate random variable called factor of safety on ground acceleration capacity (PGA for failure) above design basis PGA. This factor of safety (F) is defined as follows (Kennedy and Ravindra, 1984). A=FAdesign
(6)
here, A, is the PGA for failure; Adesign, design basis PGA, for NPPs usually it is the PGA value associated with the safe shutdown earthquake (SSE). For structures, the factor of safety can be modelled as a product of three random variables (Kennedy and Ravindra, 1984): F = FSFmFRS
(7)
here, FS, is the strength factor; Fm, is the inelastic energy absorption factor; FRS, the structural response factor. The strength factor (FS) represents the ratio of ultimate strength, which is generally corresponding to the strength at loss-of function to the stress,
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calculated for design basis PGA. It may be mentioned that, while calculating the strength factor, the non-seismic portion of the total load should be subtracted from the ultimate strength. The inelastic energy absorption factor (Fm) may be evaluated by using the ductility modified response spectra to determine the deamplification effects resulting from the inelastic energy absorption (Newmark and Hall, 1982; Chopra, 1998). The deamplification effect is related to the ductility ratio and the structural damping available within the structure. The structure response factor (FRS) is modelled as a product of factors influencing the response variability as follows (Kennedy and Ravindra, 1984). FRS = FSSFSDFMFMCFSCFDSFSSI
(8)
here, FSS, is the response spectral shape factor for variability in ground motion response spectra; FSD, is the damping factor representing the variability of structural response due to difference between actual damping and design damping; FM, is the modelling factor accounting for the uncertainty in structural response due to modelling assumptions; FMS, is the mode combination factor accounting for the variability of structural response due to method of combination of various modes of vibrations; FSC, is the spatial combination factor accounting for the variability of structural response due to the method of combination of responses due to different components of excitation; FDS, is the depth factor accounting for the change in input motion at foundation depth and FSS, is the factor accounting for the effects of soil–structure-interaction. It may be mentioned here that, the various parameters which could affect the seismic safety factor ‘F’ are assumed to be log-normally distributed (Kennedy et al., 1980). Therefore, ‘A’ in Eq. (6) is also a random variable following the same distribution, as does ‘F’. The variability of ‘A’ (or ‘F’) is due to random variation, as well as due to uncertainties, because of lack of complete and accurate knowledge. Thus, it is evident that the conditional probability of failure at a given PGA level can not be predicted with absolute certainty but rather only with an associated prob-
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ability of exceedence (or probability of non-exceedence) as given by Eqs. (2)– (4). The median value of ‘A’ and its logarithmic S.D. are related to those of ‘F’ through Eq. (6). The median value of ‘F’ is the product of median values of all its constituents as given by Eqs. (7) and (8) and the logarithmic S.D. of ‘F’ (iF) is the SRSS of the logarithmic S.D.s of all its constituents (Kennedy and Ravindra, 1984). The iF is given as follows: iF = i 2S +i 2m +i 2SS + i 2SD + i 2M + i 2MC + i 2SC + i 2DS +i 2SSI
(9) here, ‘iF’ is further divided into random variability ‘iR’ and uncertainty ‘iU’.
3. A case study
3.1. Structural features The present case study has been carried out for an elevated water-retaining structure for which the structural features are shown in Fig. 1. The structure in question is spherical in shape enclosing a central cylindrical access shaft inside and filled with water (up to the height equal to 0.15 m below the height of central cylindrical access shaft) and supported by hollow supporting concrete tower on an embedded hollow circular raft foundation. The nominal values of the relevant input data for the tank structure are given in Table 1(a), while the nominal normalised ground motion response spectral shape values at 7% structural damping are given in Table 1(b).
3.2. Mathematical modelling The structure has been modelled as an assemblage of various elements like beam, mass and spring based on the finite element technique. The hollow circular base raft, the hollow supporting concrete tower, the cupola, the spherical tank and the central cylindrical access shaft were modelled as 3-D beam elements based on conventional structural mechanics approach. The
connection between the central cylindrical access shaft and the cupola was represented by spring elements, wherein the spring constants were defined for horizontal, vertical and rocking motions based on strain energy equivalence. The lumped mass idealisation was adopted to model the mass of the structure, in which the lumped mass at any node of the discretised structure was determined from the portion of the weight that can be assigned to the node. The filled up soil around the hollow concrete tower and the bottom foundation soil below the base raft have been modelled in the form of frequency independent soil springs derived from linear elastic half space theory (ASCE, 1995). The hydrodynamic effects of water contained within the tank, have been modelled as per reference (USAEC, 1963) and accordingly, the calculated horizontal masses (i.e. impulsive mass and sloshing mass with associated horizontal spring constant) and vertical mass of water were attached at appropriate nodes of the finite element model of the structure. Fig. 2 shows the finite element model of the present structure adopted in the case study.
3.3. Parameters for 6ariation For the present case study, the variable parameters that can affect the seismic structural response can be broadly classified into three categories: (i) variable parameters for material strength, (ii) variable parameter for inelastic energy absorption and (iii) variable parameters for seismic structural response.
3.3.1. Parameters for material strength The lack of understanding of structural material properties, such as material strengths and their deviations from the anticipated design material strengths may affect the accurate estimation of the structural capacities, and this may, in turn, affect the evaluation of seismic structural margins significantly. Therefore, following parameters were considered for variation in the present case study under this category. (i) Material strengths of ‘Class-A’ and ‘Class-B’ concrete. (ii) Material strength for reinforcing steel.
(a) (i) Grade of ‘Class-A’ concrete (ii) Grade of ‘Class-B’ concrete (iii) Grade of reinforcing steel (iv) Effective prestress available in the spherical Tank between elevations 42.9 and 57.0 m (v) Poisson’s ratio for ‘Class-A’ and ‘Class-B’ concrete (vi) Unit weight of ‘Class-A’ and ‘Class-B’ concrete (vii) Ductility ratio available in the structure (viii) Structural damping for the structure (ix) Shear wave velocity for side soil (x) Shear wave velocity for bottom foundation soil (xi) Total water height in the tank (b) Time period (s) RSSV
M35 (Fck =35 MPa) M20 (Fck =20 MPa) Fe250 (Fy =250 MPa) 21 950 kN 0.15 25 kN m−3 1.0 7% 700 m s−1 5000 m s−1 13.2 m 0.0 1.000
0.1 2.145
0.2 2.715
0.4 2.190
0.6 1.620
0.8 1.230
1.0 1.095
1.5 0.860
2.0 0.665
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Table 1 Nominal values of the relevant input data for the present tank structure and nominal normalised ground motion response spectral shape values at a structural damping of 7%
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(iii) Effective prestress available in ‘Class-A’ concrete in spherical tank portion of the structure between elevations 42.9 and 57.0 m.
The Bureau of Indian Standards (BIS) (IS 432, 1982; IS 456, 1978) specify the material strength requirements for concrete and reinforcing steel in terms of characteristic strength and S.D., the for-
Fig. 1. Structural features of the elevated water retaining structure.
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Fig. 2. Finite element model of the elevated water retaining structure.
mer being a reference value preferred for specifications and design. Characteristic strength is that value of the strength below which not more than 5% of the test results are expected to fall and it has a bearing on the uncertainties in establishing the intended strength of the structure. The material strength in actual structure may be somewhat
different than the characteristic strength, this being mostly due to the inherent variability of the material properties arising in the manufacture and errors or deviations arising in the construction. The deviations in concrete strength may arise due to errors in the mix proportions, presence of impurities, inadequate compaction or curing and
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construction faults. Therefore, considering the inherent variability of concrete strength during the production stage, it becomes necessary to design the concrete mix to have a target mean strength, which is higher than the characteristic strength. However, during the testing of samples for strength, some samples may have the strength which is less than the characteristic strength, these samples shall also be deemed to comply with the strength requirements if they fulfil the strength requirements stipulated in the BIS (IS 456, 1978). Considering these, the material strength for ‘Class-A’ concrete has been varied from 28 to 43 MPa and that for ‘Class-B’ concrete has been varied from 16 to 26 MPa. The deviations in reinforcing steel strength may arise largely due to the manufacturing defects such as the deviations from the nominal diameter and the metallurgical composition of the reinforcing steel bars. Sometimes inaccurate positioning of the reinforcing bars may result in deviation of anticipated strength of the individual members. However, BIS (IS 432, 1982), stipulates that, during the testing of samples for strength, no sample shall have the strength, which is less than the characteristic strength. Considering this, the material strength of the reinforcing steel has been varied from 250 to 325 Mpa by assuming a coefficient of variance of 10% for the material strength. The effective prestress in concrete, undergoes a gradual reduction with time from the stage of transfer due to various causes. This is generally referred to as loss of prestress which depends on several factors, such as the properties of the concrete and steel, method of curing, degree of prestress and process of prestressing. Therefore, it is difficult to generalise the exact amount of loss of prestress. However, based on the available data (Raju, 1990), typical values of the loss of prestress that could be encountered under normal conditions were considered and the effective prestress available in the ‘Class-A’ concrete in spherical tank portion of tank between elevations 42.9 and 57.0 m has been varied from 21 950 to 16 460 kN, which would be equivalent to 80– 60% of the initial prestressing.
3.3.2. Parameter for inelastic energy absorption An earthquake represents a source of limited energy, and therefore, it is normally expected that in the event of an earthquake occurrence, the structure must display reasonably good energy absorption characteristics by undergoing large deformations even beyond the yield without loosing much of its load carrying capacity and at the same time it should not fail to serve the various process features to which it is supposed to cater for. This ideal situation would require that the structure should be sufficiently ductile, so that it can deform inelastically beyond the yield and in doing so will redistribute the excess load to elastic parts of the structure. It may be noted that, in reinforced concrete structures, the ductility is largely governed by various factors viz. material properties of concrete and reinforcing steel, confinement of concrete, amount of reinforcing steel provided and the rate of loading and could be improved upon by proper structural detailing and control of construction procedures. Therefore, it may be mentioned that, the ductility available within the structure is greatly influenced by the variability associated with the construction materials (i.e. the concrete and the reinforcing steel) and the structural reinforcement detailing aspects. In the present case study, the only parameter considered under this category was ductility ratio available within the structure and in the absence of any available experimental data to establish the exact ductility ratio for the present structure, it has been varied from 1.0 to 1.75 to maintain reasonable conservatism in the seismic response analysis. No non-linear analyses were performed in the present study instead, the effect of deamplification resulting from inelastic dissipation was taken into account by using the ductility modified response spectra (Newmark and Hall, 1982; Chopra, 1998), which was derived from the elastic response spectra corresponding to median damping value and the ductility ratio. 3.3.3. Parameters for structural response For satisfactory performance in the event of an earthquake occurrence, generally the structures are analysed and designed by using specific deterministic response parameters for the structures,
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which may often be highly conservative. Since, many of these response parameters are random in nature with wide variability, most oftenly associated with them, the actual structural response may substantially differ from the structural response, calculated at the time of designing the structures. Therefore, the following variable parameters that may affect the calculated seismic response of the structures, were considered in the present case study under this category. (i) Spectral shape, representing the variability in ground motion and associated ground response spectra. (ii) Damping, representing the variability in response due to difference between actual damping and design damping. (iii) Soil–structure interaction, representing the variability in response due to variation in soil parameters for side soil and foundation soil. (iv) Fluid– structure interaction, representing the variability in response due to variation in the total height of water available inside the tank. For the aseismic design of structures as well as for the seismic re-qualification of the existing structures, the most important consideration is to derive a site-specific design basis response spectrum and at the same time the shape of the spectra is largely governed by the local geological and tectonic features. Therefore, the derivation of suitable design basis response spectral shape should be done by considering a large number of records having earthquake parameters in the range of interest and for better compatibility, these data should be drawn from sites having similar geological and tectonic conditions. The statistical combination of response spectra of number of individual records will then result in the site-specific design basis response spectra. However, it may be mentioned that, the mean (M) and standard deviation (S.D.) (|) of spectral values at all frequencies reflect the variations in the source-transmission-path site characteristics and help in assigning statistical confidence limits with specified spectral values to achieve an acceptable level of conservatism in the analyses. Generally, (M +|) response spectral shapes having a
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confidence level of 84% are used for aseismic design of structures for regions with low seismic risk. In the present case study, the site-specific design basis peak response spectral shape values corresponding to the time period of 0.2 s have been varied from 1.5395 to 3.8905, to represent the wider range of the variation in seismo-tectonic status of the region. For the vary purpose of seismic re-qualification of existing structures, it could be ideal to determine experimentally the important properties of the structures including its damping, but this is rarely done, generally, because of the unavailability of the experimental set-up at the actual structure site and due to lack of budget and time. Therefore, for an existing structure to be re-evaluated for its seismic safety, the selection of damping ratio is generally based on the available data on damping extracted from recorded earthquake motions of many structures of different types and different construction materials. However, in the present case study, due to unavailability of experimental data on damping, the damping ratios for the structure were taken from available data set (Newmark and Hall, 1982), and varied from 4 to 10%. It is widely recognised that the behaviour of a structure during an earthquake may be significantly affected by the nature of the foundation and surrounding soil media. Various authors have proposed various models of soil– structure interaction and simplified criteria to account for or ignore the soil– structure interaction. However, it has been observed that various criteria to ignore soil–structure interaction and carry out analysis with a fixed-base assumption depend on case specific parameters and as such the conservatism of fixed-base analysis can not be generalised (Patnaik et al., 1995; Ghosh et al., 1997). In the present case study, the uncertainties in soil–structure interaction are accounted for by considering the variation in shear wave velocity of soil and in the absence of adequate soil investigation data the shear wave velocity for side soil has been varied from 350 to 1400 m s − 1, while that for foundation soil has been varied from 1800 to 5000 m s − 1.
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One of the practical problems that generally occurs in the design of liquid-retaining structures, is the interaction between the contained liquid and the structure itself during the vibrations under the influence of seismic ground motion. The resulting horizontal hydrodynamic pressures, i.e. impulsive and sloshing due to fluid – structure interaction effects depend on the characteristics of the excitation, the properties and the level of the contained liquid and the geometric and physical properties of the liquid-retaining structure itself (USAEC, 1963). The fluid height largely affects the distribution of the impulsive and sloshing components of the hydrodynamic pressures along the height of the structure as well as the availability of effective fluid mass for the vertical seismic ground motion, and this, in turn, may affect the seismic structural response significantly. Consider-
ing these, the effective total water height in the present structure has been varied from 13.2 to 7.9 m in the present study. Table 2(a) summarises the range of variation of the relevant input data for the tank structure about their nominal values while the Table 2(b) gives the range of variation of the normalised ground motion response spectral shape values about their nominal values.
3.4. Failure mode for the present case study The first step in developing a fragility description, is to identify the failure mode associated with the fragility curves and the consequences of the failure mode. Identification of the most likely failure modes, caused by the seismic event, is generally governed by the experience and judge-
Table 2 Range of variation Category
Parameter
Rele6ant input data for the present tank structure (a) (a) Material strength (i) ‘Class-A’ concrete (ii) ‘Class-B’ concrete (iii) Reinforcing steel (iv) Effective prestress in the spherical tank portion between elevation 42.9 and 57.0 m (b) Inelastic energy (i) Ductility ratio available with in the absorption structure (c) Structural response (i) Structural damping (ii) Shear wave velocity for the side soil (iii) Shear wave velocity for the bottom foundation soil (iv) Total water height in the tank Normalised ground motion response spectral shape 6alues at a structural damping of 7% (b) Time period (s) RSSV 0.0 1.000 (constant) 0.1 1.4055–3.1680 0.2 1.5395–3.8905 0.4 1.1545–3.2255 0.6 0.7805–2.4595 0.8 0.5365–1.9235 1.0 0.4280–1.7620 1.5 0.2380–1.4820 2.0 0.1355–1.2215 Category: structural response; parameter: response spectral shape.
Range of variation
28–43 MPa 16–26 MPa 250–325 MPa 21 950–16 460 KN 1.0–1.75 4–10% 350–1400 m s−1. 1800–5000 m s−1. 13.2–7.9 m
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ment of the structural analyst engaged in generating the fragility curves. In general, the structures are considered to fail functionally when the degree of distress in the structures due to seismic event achieves a certain level beyond which, it would be reasonable to assume that the appropriate and designated functioning of the safety-related equipment attached to the structures, could be interfered substantially. However, this degree of distress is very judgmental and must reflect the location of the structural distress and the attached safety-related equipment. It may be further noted that the failure is characterised by a cumulative distribution function which describes the probability that failure has occurred, given a value of loading and in the context of seismic fragility this loading could be represented in terms of PGA, depending on the structure and the failure mode under consideration. In the present case study, it would be reasonable to consider that, under seismic excitation, the present elevated water-retaining structure essentially behaves like a cantilever beam with circular cross-section, somewhat modified by the rocking of the base raft. Therefore, it would be appropriate to assume bending of structure in the direction of seismic ground motion as a dominant mode of failure and this will result in the linear variation of the axial bending stresses across the cross-section as long as the response of the structure stays in the elastic range. The structure was considered to fail when the axial bending stress due to seismic event achieves the permissible axial bending stress at the locations of interest. For ‘Class-A’ concrete, the compressive axial bending stress due to seismic event shall be resisted by the concrete and reinforcing steel together, while the tensile axial bending stress shall be resisted by the prestressing tendons and the reinforcing steel together. It may be noted that, for ‘Class-A’ concrete, only after the tensile axial bending stress due to seismic event exceeds the available prestress, will the bonded reinforcing steel experience the load. For ‘Class-B’ concrete, the compressive axial bending stress shall be resisted by the concrete and reinforcing steel together, while the tensile
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axial bending stress shall be resisted by the reinforcing steel alone. The typical axial bending stress distribution across the cross-section due to seismic ground motion in one of the direction is shown in Fig. 3. Table 3 gives the typical values of permissible axial bending stresses at different locations considered in the present case study, for the nominal values of the input data. Here, it may be mentioned that, while calculating the permissible axial bending stresses at different locations the non-seismic portion of the total load acting on the structure has been subtracted from the available capacities, therefore, the values mentioned in the Table 3 represent the portion of axial bending stress available for the seismic load only.
3.5. Numerical analyses The present elevated water-retaining structure has been analysed by using the response spectrum technique. In the response spectrum method of analysis, linear elastic structures subjected to base excitation are analysed by solving the uncoupled equations of motion for maximum modal responses. The total response of the structure is obtained by summing up the contribution of response due to each mode. In the present case study, the seismic input includes two horizontal and the vertical components of the ground motions in combination. For the present structure the maximum response has been obtained by using CQC method for modal combination (ASCE, 1995) and the spatial combination for horizontal and vertical ground motions has been performed using the SRSS technique (ASCE, 1995). Software program COSMOS/M (COSMOS/M, 1997) was used for all analyses purposes. The analyses of the present tank structure were carried out for two conditions: (i) tank empty and (ii) tank filled with water. The PGA to failure was determined at two different locations: (i) node ‘9’ for the element ‘8’ and (ii) node ‘20’ for the element ‘19’ as shown in the Fig. 2 and these two elements were chosen, because of the maximum stress experienced by them due to given ground motion.
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Fig. 3. Typical axial bending stress distribution across the cross-section of wall.
Parametric studies were made by varying the material strengths of ‘Class-A’ concrete, ‘ClassB’ concrete and reinforcing steel, effective prestress available in ‘Class-A’ concrete in spherical tank portion, ductility ratio and damping available within the structure, normalised ground motion response spectral shapes, shear wave ve-
locity for the surrounding and foundation soil media and the total height of water in the tank about their nominal values making change only in one variable parameter at a time. It may be noted here that, variations in ductility ratio available within the structure and the total water height in the tank were considered only for
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the tank filled with water condition for the present case study. The results of the parametric studies were then used to determine the seismic safety factor (i.e. the ratio of PGA for failure to the design basis PGA) and its variability (i.e. its logarithmic mean value and logarithmic S.D.) with each of the parameters mentioned above for the bending mode of failure. Here, it may be mentioned that while calculating the variability for each of the parameters, different weights were assigned to the range of variation for the parameter considered at a time as per the probability of its occurrence, with the maximum weight assigned to the nominal value of the parameter which is the median value of the parameter. Here, while assigning the weights it was further assumed that, the variable parameters follow log-normal probability distribution function. In the absence of sufficient experimental data the logarithmic S.D. due to randomness about the median value (iR) due to individual parameters have been evaluated through a parametric study and later on the logarithmic S.D. due to uncertainties in the median values (iU) were evaluated based on the range of the iR and iU available in the literature (Kennedy and Ravindra, 1984) for the present case study. After the evaluation of variabilities due to individual parameters, they were combined to determine the overall logarithmic S.D. of the seismic safety factor due to randomness about the median value and due to the uncertainties in the median values for the structure at two different locations as mentioned above. Here it may be noted that the overall logarithmic S.D.
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of the seismic safety factor for the structure is the SRSS combination of the logarithmic S.D. of the individual parameters considered in the analyses. For the present case study Table 4(a and b), Table 5(a and b) give the calculated values of iR and iU at two different locations of the present structure (i.e. at node ‘9’ of element ‘8’ and node ‘20’ of element ‘19’). Table 6 summarises the values of median ground acceleration capacity (A( ) and composite variability (iC) at the locations of interest for the different tank conditions. In the present case study the seismic fragility of the structure in question has been evaluated considering the variation in the number of parameters as mentioned previously. Figs. 4–11 represent the seismic fragility of the present structure developed as families of conditional probability curves plotted against PGA at two different locations of interest for the different tank conditions.
4. Discussion of results and conclusions The seismic fragility curves are generated in the present case study considering the log-normal distribution of the various parameters which may be justified as an appropriate distribution, because of the fact that the statistical variation of many material properties and seismic structural response variables may be represented reasonably by this distribution. Moreover, by using the log-normal distribution for the variable parameters, the entire fragility curve and its uncertainty can be expressed by only three
Table 3 Typical values of permissible axial bending stress for the nominal values of the relevant input data Location
Node ‘9’ of element ‘8’ Node ‘20’ of element ‘19’
Tank condition
(i) Tank empty (ii) Tank filled with water (i) Tank empty (ii) Tank filled with water
Permissible axial bending stress (MPa) Compression
Tension
7.7362 4.3319 8.1658 7.0484
1.7305 5.1348 1.4438 2.5612
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Table 4 Calculated values of iR and iU for location at node ‘9’ of element ‘8’ Category
Parameter
(a) Tank empty condition (a) Material strength (i) ‘Class-A’ concrete (ii) ‘Class-B’ concrete (iii) Reinforcing steel (b) Structural response (i) Structural damping (ii) Response spectral shape (iii) Shear wave velocity for the side soil (iv) Shear wave velocity for the bottom foundation soil Overall variability (b) Tank filled with water condition (a) Material strength (i) ‘Class-A’ concrete (ii) ‘Class-B’ concrete (iii) Reinforcing steel (b) Inelastic energy (i) Ductility ratio available with in the structure absorption (c) Structural response (i) Structural damping (ii) Response spectral shape (iii) Shear wave velocity for the side soil (iv) Shear wave velocity for the bottom foundation soil (v) Total water height in the tank Overall variability
parameters namely median ground acceleration capacity (A( ), logarithmic S.D. due to randomness about the median value (iR) and logarithmic S.D. due to uncertainty in the median value (iU), which are easier to estimate. From Table 4(a and b), Table 5(a and b), it is very clear that, a substantially greater amount of variability (both iR and iU) is, because of the contribution from material strengths for ‘Class-A’ and ‘Class-B’ concrete, normalised ground motion response spectral shapes, ductility ratio available with in the structure and total water height available in the tank, therefore, the response of the present structure at the locations of interest appears to be quite sensitive to the variation of these parameters in the present case study. Similarly, it has also been observed from the same tables that, the overall variability due to uncertainty in median value (iU) is more than the overall variability due to randomness about the median value (iR) for bending compression mode of failure while the overall iU is
Bending compression
Bending tension
iR
iU
iR
iU
0.0003 0.1698 0.0054 0.0570 0.2205 0.0271 0.0285 0.2869
0.0005 0.2971 0.0094 0.0570 0.1103 0.1220 0.1284 0.3676
0.0003 0.0156 0.0300 0.0570 0.2279 0.0273 0.0285 0.2406
0.0005 0.0273 0.0525 0.0570 0.1140 0.1228 0.1284 0.2265
0.0024 0.2941 0.0095 0.0672
0.0042 0.5147 0.0167 0.1378
0.0025 0.0162 0.0104 0.0685
0.0043 0.0284 0.0183 0.1405
0.0617 0.2840 0.0349 0.0245 0.2114 0.4712
0.0617 0.1420 0.1572 0.1104 0.2114 0.6243
0.0616 0.2840 0.0350 0.0247 0.0701 0.3103
0.0616 0.1240 0.1575 0.1110 0.0701 0.2948
observed to be less than the overall iR for the bending tension mode of failure. This may be attributed to the fact that the evaluation of axial bending tension capacities are mainly governed by the contribution from reinforcing steel only and are supposed to be free from the uncertainty associated with the material strength parameters for concrete. From Table 6, it can reasonably be inferred that, in the present case study, lesser composite variability is associated with the tank empty condition as compared with the tank filled with water condition which reflects the significant contribution from the total water height towards the composite variability for the seismic structural response. Figs. 4–11 represent the seismic fragility of the structure at the locations of interest for different modes of bending failure, different tank conditions and different levels of confidence. In the present case study, the overall logarithmic S.D. due to randomness about the median value (iR) at different locations and for differ-
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Table 5 Calculated values of iR and iU for location at node ‘20’ of element ‘19’ Category
Parameter
(a) Tank empty condition (a) Material (i) ‘Class-A’ concrete strength (ii) ‘Class-B’ concrete (iii) Reinforcing steel (iv) Effective prestress in the spherical tank portion between elevations 42.9 and 57.0 m (b) Structural (i) Structural damping response (ii) Response spectral shape (iii) Shear wave velocity for the side soil (iv) Shear wave velocity for the bottom foundation soil Overall variability
(b) Tank filled with water condition (a) Material (i) ‘Class-A’ concrete strength (ii) ‘Class-B’ concrete (iii) reinforcing steel (iv) Effective prestress in the spherical tank portion between elevations 42.9 and 57.0 m (b) Inelastic (i) Ductility ratio available with in the structure energy absorption (c) Structural (i) Structural damping response (ii) Response spectral shape (iii) Shear wave velocity for the side soil (iv) Shear wave velocity for the bottom foundation soil (v) Total water height in the tank Overall variability
Bending compression
Bending tension
PR
PU
PR
PU
0.1490
0.2607
0.0076
0.0133
0.0070 0.0031 0.0029
0.0122 0.0055 0.0051
0.0070 0.0187 0.0179
0.0122 0.0328 0.0313
0.0605
0.0605
0.0606
0.0606
0.2445 0.0130 0.0020 0.2930
0.1222 0.0583 0.0091 0.3004
0.2444 0.0130 0.0021 0.2537
0.1222 0.0580 0.0094 0.1563
0.1675
0.2930
0.0030
0.0052
0.0046 0.0036 0.0033
0.0080 0.0063 0.0059
0.0045 0.0106 0.0098
0.0078 0.0185 0.0171
0.1190
0.2440
0.1190
0.2439
0.0609
0.0609
0.0609
0.0609
0.3030 0.0049 0.0009 0.1765 0.4111
0.1515 0.0221 0.0040 0.1765 0.4515
0.3030 0.0049 0.0010 0.1167 0.3515
0.1515 0.0221 0.0045 0.1167 0.3178
Table 6 Median ground acceleration capacity (A( ) and composite variability (iC) at different locations of interest for the present tank structure Location
Node ‘9’ of element ‘8’ Node ‘20’ of element ‘19’
Tank condition
Tank Tank Tank Tank
empty filled with water empty filled with water
Bending compression
Bending tension
A( (g)
iC
A( (g)
iC
3.4391 0.3102 1.5406 0.3885
0.4663 0.7822 0.4196 0.6106
0.7693 0.3676 0.2724 0.1412
0.3305 0.4280 0.2980 0.4739
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Fig. 4. Seismic fragility curves for location at node ‘9’ of element ‘8’.
Fig. 5. Seismic fragility curves for location at node ‘9’ of element ‘8’.
ent tank conditions, was evaluated through parametric studies and later on the values for overall logarithmic S.D. due to uncertainty in the median value (iU) are evaluated based on the data available in the literature (Kennedy and Ravindra, 1984).
However, it may be mentioned that, the factors iR and iU should be evaluated separately for the structure, and this, in turn, requires performing a lot of experiments and analyses in order to find the variations of different variable parameters.
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In the present case study, the seismic response of the structure at the locations of interest has been found to be quite sensitive to the randomness
269
and uncertainty associated with the material strength of concrete, normalised ground motion response spectral shapes, ductility ratio variable
Fig. 6. Seismic fragility curves for location at node ‘9’ of element ‘8’.
Fig. 7. Seismic fragility curves for location at node ‘9’ of element ‘8’.
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Fig. 8. Seismic fragility curves for location at node ‘20’ for element ‘19’.
Fig. 9. Seismic fragility curves for location at node ‘20’ of element ‘19’.
available within the structure and the total water height in the tank. By carrying out extensive experiments combined with the analysis work to ascertain the variation in the various variable
parameters, which may contribute towards the randomness, and uncertainty in the seismic structural response, the fragility evaluation may be carried out in a more realistic manner.
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271
Fig. 10. Seismic fragility curves for location at node ‘20’ of element ‘19’.
Fig. 11. Seismic fragility curves for location at node ‘20’ of element ‘19’.
Appendix A. Nomenclature a A A(
any value of ground acceleration peak ground acceleration, ground acceleration capacity, a random variable median ground acceleration capacity of a structure
Adesign ‘Class-A’ ‘Class-B’ DS
design basis PGA value prestressed concrete (nominal value of FCK = 35 MPa) reinforced concrete (nominal value of FCK = 20 MPa) random variable accounting for the change in input motion at foundation depth
272
F F( ) FCK FY M MC
P Pf P %f
Q RS RSSV S SC
SD
SRSS SS
SSI iC iF iR iU
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factor of safety above design basis PGA factor of safety for the variable specified in the subscript characteristic strength for concrete characteristic yield strength for steel random variable reflecting the variability due to modelling assumptions random variable reflecting the variability due to method of combination of various modes of vibrations exceedence probability level conditional frequency of failure at any exceedence probability level (includes both randomness and uncertainty) conditional frequency of failure at any non-exceedence probability level (includes both randomness and uncertainty) non-exceedence probability level structural response response spectral shape values strength of a structure random variable accounting for the variability due to method of earthquake component combination random variable reflecting the variability due to difference in actual and design structural damping square root of sum of squares random variable reflecting the variability in the response spectral shapes used in the design random variable reflecting the variability due to the effects of soil-structure interaction composite logarithmic S.D. logarithmic S.D. of ‘F’ logarithmic S.D. representing the inherent randomness in the variable logarithmic S.D. representing the uncertainty in the variable
v ( )
ductility ratio standard Gaussian cumulative distribution function −1( ) inverse of standard Gaussian cumulative distribution function
References ASCE Standards, 1995. Seismic Analysis of Safety-Related Nuclear Structures and Commentary, Fourth Revision, American Society of Civil Engineers, Washington, pp. 3.18 – 3.28. Chopra, A.K., 1998. Dynamics of Structures, Theory and Applications to Earthquake Engineering. Prentice-Hall, New Delhi, India, pp. 269 – 275. COSMOS/M 2.0, 1997. A Complete Finite Element Analysis System, Structural Research & Analysis Corporation, Los Angeles. Ghosh, A.K., Agrawal, M.K., Patnaik, R., Bhargava, K., 1997. Effect of soil parameters on the seismic response of a 3-D frame structure. Transaction of the 14th International Conference on Structural Mechanics in Reactor Technology, Lyon, France, pp. 331 – 338. IS: 432 (Part-I), 1982. Indian Standard Specification for Mild Steel and Medium Tensile Steel Bars and Hard-Drawn Steel Wire for Concrete Reinforcement, Third revision, Bureau of Indian Standards, New Delhi, pp. 3 – 9. IS: 456, 1978. Indian Standard Code of Practice for Plain and Reinforced concrete, Third Revision, Bureau of Indian Standards, New Delhi, pp. 40 – 45. Kennedy, R.P., Ravindra, M.K., 1984. Seismic fragilities for nuclear power plant risk studies. Nucl. Eng. Des. 79, 47 – 68. Kennedy, R.P., Cornell, C.A., Campbell, R.D., Kaplan, S., Perla, H.F., 1980. Probabilistic seismic safety study of an existing nuclear power plant. Nucl. Eng. Des. 59, 315 –338. Newmark, N.M., Hall, W.J., 1982. Earthquake Spectra and Design. Earthquake Engineering Research Institute, Berkeley, CA, pp. 29 – 54. Patnaik, R., Bhargava, K., Ghosh, A.K., Agrawal, M.K., 1995. Effect of foundation conditions on the seismic response of a space frame. Transactions of the 13th International Conference on Structural Mechanics in Reactor Technology, Porto Alegre, Brazil, pp. 163 – 168. Raju, N.K., 1990. Prestressed Concrete, second ed. Tata McGraw-Hill, New Delhi, pp. 73 – 79. USAEC, 1963. Nuclear Reactors and Earthquakes (TID7024), US Atomic Energy Commission, Washington, pp. 185 – 188.
Evaluation of seismic fragility of structures —a case study Kapilesh Bhargava a,*, A.K. Ghosh b, M.K. Agrawal b, R. Patnaik a, S. Ramanujam a, H.S. Kushwaha b a
Architecture and Ci6il Engineering Di6ision, Bhabha Atomic Research Centre, Mumbai 400 085, India b Reactor Safety Di6ision, Bhabha Atomic Research Centre, Mumbai, India
Abstract The present paper attempts to evaluate the seismic fragility for a typical elevated water-retaining structure. The structure is analysed for two cases: (i) empty tank; and (ii) tank filled with water. The various parameters that could affect the seismic structural response include material strength of concrete and reinforcing steel, effective prestress available in the tank, ductility ratio and structural damping available within the structure, normalised ground motion response spectral shape, foundation and surrounding soil parameters and the total height of water available in the tank. Based on this case study, the seismic fragility of the structure is developed. The results are presented as families of conditional probability curves plotted against peak ground acceleration (PGA) at two critical locations. The procedure adopted, incorporates the various randomness and uncertainty associated with the parameters under consideration. © 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The safety of a nuclear power plant (NPP) depends upon a number of factors — intrinsic and external to the plant. Seismic ground motion is an important consideration in evaluating the safety of the plant or, alternatively, the risk associated with it. The various randomness and uncertainties associated with the occurrence of earthquakes and the consequences of their effects on the NPP components and structures call for a probabilistic risk assessment (PRA). Probabilistic seismic risk analysis (PSRA) have been or being carried out * Corresponding author. Tel.: + 91-22-5505050x22494; fax: + 91-22-5505151/5519613. E-mail address: kapil – [email protected] (K. Bhargava).
for various NPPs all over the world (Kennedy et al., 1980). Many of the existing nuclear facilities were designed and constructed during the time when seismic design procedure was evolving. Mostly, these plants were designed using the seismic coefficient method. In recent years, an increasing knowledge in the geoscience has led to a better understanding of seismic ground motion. It is also known that older plants were designed to seismic criteria that were less rigorous than those used for recent plants. In addition to this, utilities and regulatory agencies are making continuous efforts to modernise and update the structural safety due to increase in evidence of seismic hazard not envisaged earlier. The primary objective of the seismic review of the older plants is to make an overall safety assessment of the seismic capabili-
0029-5493/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 9 - 5 4 9 3 ( 0 1 ) 0 0 4 9 1 - 5
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ties of these plants, and, if necessary, to upgrade them to ensure that these plants can be safely shut down in the event of a design basis earthquake. For demonstrating the overall safety of the plant, both regulatory agencies and plant designers together have to develop a review basis ground motion (RBGM) or seismic margin earthquake (SME). A general definition of seismic margin is expressed in terms the earthquake peak ground acceleration (PGA) that compromises plant safety, specifically leading to melting of reactor core. Seismic margin assessment (SMA) involves the review of the capacity of the plant vis-a`-vis the target earthquake input (RBGM or SME) selected for review. The objectives are then to show whether the plant can withstand the effects of this SME with high confidence and to identify seismic vulnerabilities. This can be fulfilled using the results and insights obtained from PSRA, the data on actual performance of structures and equipment in actual earthquakes and tests. The key elements of PSRA comprise the evaluation of the following parameters considering the variations due to their randomness and uncertainties. (i) seismic hazard analysis at the site. (ii) Response of plant systems and structures. (iii) Seismic fragility evaluation. (iv) Systems or accident sequence analysis. (v) Evaluation of core damage frequency. It may be mentioned here that the items (ii) and (iii) are intrinsically related to each other. The seismic PRA would provide answers regarding the seismic capacities of the systems, structures and components of the plant. The large uncertainties in the seismic hazard curves make decisions regarding seismic capacity difficult whereas, the SMA review is mostly focussed on the few structures and components in the plant whose failure would lead to severe core damage. The output of SMA review is an estimation of the plant seismic capacity whereas, the seismic PRA provides estimation of seismic risks of core damage and adverse public effects. The present paper is concerned with the response of an existing elevated water-retaining structure to seismic excitation and the evaluation of seismic fragilities at two critical locations for bending mode of failure. The methodology adopted in the present
case study would be useful in the seismic reassessment of existing structures.
2. Seismic fragility and fragility model The seismic fragility of a structure may be defined as the conditional probability of its failure for a given value of the seismic input parameter e.g. PGA. The failure could be defined in terms of a limiting value of stress and/or displacement. Estimation of this PGA value, also known as ground acceleration capacity of the structure, is generally carried out by using the information available on the structural design bases, structural configurations, material properties and seismic structural response evaluated at the design stage of the structure. It may be mentioned that, there are many sources of randomness and uncertainty, that may affect the accurate estimation of the structural capacity, and this, may, in turn, affect the ground acceleration capacity for the structure significantly. Therefore, the seismic fragility is generally described by means of a family of fragility curves along with a probability value assigned to each curve to reflect the confidence level associated with the estimation of fragility. The entire fragility family for a structure corresponding to a particular mode of failure can be expressed in terms of the median ground acceleration capacity (A( ) and two random variables (mR and mU) as follows (Kennedy et al., 1980). A= A( mRmU
(1)
Here, mR and mU are the random variables with unit median values and represent the inherent randomness about the median value and the uncertainty in the median value respectively. Both mR and mU are assumed to be log-normally distributed with logarithmic standard deviations (S.D.) iR and iU respectively. The uncertainty associated with the fragility evaluation is generally expressed in terms of a range of failure probabilities for a given value of ground acceleration. The probability (P) that the conditional failure frequency (Pf) exceeds a certain specified value, P %f for a ground acceleration ‘a’ is given by (Kennedy et al., 1980).
P
n
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PfP%f ln{a/A( exp[iR = iU a
−1
n
(P%f)]}
(2)
Alternatively, the conditional failure frequency P %f at a non-exceedence probability (Q) can be expressed as (Kennedy and Ravindra, 1984): P%f =
ln(a/A( ) + iU − 1(Q) iR
Q= 1 − P
n
(3) (4)
here, (x), is the standard Gaussian cumulative distribution function; − 1 (P), the inverse of the standard Gaussian cumulative distribution function. For the purpose of displaying the fragility curves, the non-exceedence probability level ‘Q’ is utilised. In some applications, the composite variability iC is used which is defined by the following (Kennedy et al., 1980): iC = i 2R +i 2U
(5)
It may be mentioned here that, the use of iC and A( provides a single best estimate fragility curve which does not explicitly separate out uncertainty from underlying randomness. In the estimation of fragility parameters, it is convenient to evolve an intermediate random variable called factor of safety on ground acceleration capacity (PGA for failure) above design basis PGA. This factor of safety (F) is defined as follows (Kennedy and Ravindra, 1984). A=FAdesign
(6)
here, A, is the PGA for failure; Adesign, design basis PGA, for NPPs usually it is the PGA value associated with the safe shutdown earthquake (SSE). For structures, the factor of safety can be modelled as a product of three random variables (Kennedy and Ravindra, 1984): F = FSFmFRS
(7)
here, FS, is the strength factor; Fm, is the inelastic energy absorption factor; FRS, the structural response factor. The strength factor (FS) represents the ratio of ultimate strength, which is generally corresponding to the strength at loss-of function to the stress,
255
calculated for design basis PGA. It may be mentioned that, while calculating the strength factor, the non-seismic portion of the total load should be subtracted from the ultimate strength. The inelastic energy absorption factor (Fm) may be evaluated by using the ductility modified response spectra to determine the deamplification effects resulting from the inelastic energy absorption (Newmark and Hall, 1982; Chopra, 1998). The deamplification effect is related to the ductility ratio and the structural damping available within the structure. The structure response factor (FRS) is modelled as a product of factors influencing the response variability as follows (Kennedy and Ravindra, 1984). FRS = FSSFSDFMFMCFSCFDSFSSI
(8)
here, FSS, is the response spectral shape factor for variability in ground motion response spectra; FSD, is the damping factor representing the variability of structural response due to difference between actual damping and design damping; FM, is the modelling factor accounting for the uncertainty in structural response due to modelling assumptions; FMS, is the mode combination factor accounting for the variability of structural response due to method of combination of various modes of vibrations; FSC, is the spatial combination factor accounting for the variability of structural response due to the method of combination of responses due to different components of excitation; FDS, is the depth factor accounting for the change in input motion at foundation depth and FSS, is the factor accounting for the effects of soil–structure-interaction. It may be mentioned here that, the various parameters which could affect the seismic safety factor ‘F’ are assumed to be log-normally distributed (Kennedy et al., 1980). Therefore, ‘A’ in Eq. (6) is also a random variable following the same distribution, as does ‘F’. The variability of ‘A’ (or ‘F’) is due to random variation, as well as due to uncertainties, because of lack of complete and accurate knowledge. Thus, it is evident that the conditional probability of failure at a given PGA level can not be predicted with absolute certainty but rather only with an associated prob-
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ability of exceedence (or probability of non-exceedence) as given by Eqs. (2)– (4). The median value of ‘A’ and its logarithmic S.D. are related to those of ‘F’ through Eq. (6). The median value of ‘F’ is the product of median values of all its constituents as given by Eqs. (7) and (8) and the logarithmic S.D. of ‘F’ (iF) is the SRSS of the logarithmic S.D.s of all its constituents (Kennedy and Ravindra, 1984). The iF is given as follows: iF = i 2S +i 2m +i 2SS + i 2SD + i 2M + i 2MC + i 2SC + i 2DS +i 2SSI
(9) here, ‘iF’ is further divided into random variability ‘iR’ and uncertainty ‘iU’.
3. A case study
3.1. Structural features The present case study has been carried out for an elevated water-retaining structure for which the structural features are shown in Fig. 1. The structure in question is spherical in shape enclosing a central cylindrical access shaft inside and filled with water (up to the height equal to 0.15 m below the height of central cylindrical access shaft) and supported by hollow supporting concrete tower on an embedded hollow circular raft foundation. The nominal values of the relevant input data for the tank structure are given in Table 1(a), while the nominal normalised ground motion response spectral shape values at 7% structural damping are given in Table 1(b).
3.2. Mathematical modelling The structure has been modelled as an assemblage of various elements like beam, mass and spring based on the finite element technique. The hollow circular base raft, the hollow supporting concrete tower, the cupola, the spherical tank and the central cylindrical access shaft were modelled as 3-D beam elements based on conventional structural mechanics approach. The
connection between the central cylindrical access shaft and the cupola was represented by spring elements, wherein the spring constants were defined for horizontal, vertical and rocking motions based on strain energy equivalence. The lumped mass idealisation was adopted to model the mass of the structure, in which the lumped mass at any node of the discretised structure was determined from the portion of the weight that can be assigned to the node. The filled up soil around the hollow concrete tower and the bottom foundation soil below the base raft have been modelled in the form of frequency independent soil springs derived from linear elastic half space theory (ASCE, 1995). The hydrodynamic effects of water contained within the tank, have been modelled as per reference (USAEC, 1963) and accordingly, the calculated horizontal masses (i.e. impulsive mass and sloshing mass with associated horizontal spring constant) and vertical mass of water were attached at appropriate nodes of the finite element model of the structure. Fig. 2 shows the finite element model of the present structure adopted in the case study.
3.3. Parameters for 6ariation For the present case study, the variable parameters that can affect the seismic structural response can be broadly classified into three categories: (i) variable parameters for material strength, (ii) variable parameter for inelastic energy absorption and (iii) variable parameters for seismic structural response.
3.3.1. Parameters for material strength The lack of understanding of structural material properties, such as material strengths and their deviations from the anticipated design material strengths may affect the accurate estimation of the structural capacities, and this may, in turn, affect the evaluation of seismic structural margins significantly. Therefore, following parameters were considered for variation in the present case study under this category. (i) Material strengths of ‘Class-A’ and ‘Class-B’ concrete. (ii) Material strength for reinforcing steel.
(a) (i) Grade of ‘Class-A’ concrete (ii) Grade of ‘Class-B’ concrete (iii) Grade of reinforcing steel (iv) Effective prestress available in the spherical Tank between elevations 42.9 and 57.0 m (v) Poisson’s ratio for ‘Class-A’ and ‘Class-B’ concrete (vi) Unit weight of ‘Class-A’ and ‘Class-B’ concrete (vii) Ductility ratio available in the structure (viii) Structural damping for the structure (ix) Shear wave velocity for side soil (x) Shear wave velocity for bottom foundation soil (xi) Total water height in the tank (b) Time period (s) RSSV
M35 (Fck =35 MPa) M20 (Fck =20 MPa) Fe250 (Fy =250 MPa) 21 950 kN 0.15 25 kN m−3 1.0 7% 700 m s−1 5000 m s−1 13.2 m 0.0 1.000
0.1 2.145
0.2 2.715
0.4 2.190
0.6 1.620
0.8 1.230
1.0 1.095
1.5 0.860
2.0 0.665
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Table 1 Nominal values of the relevant input data for the present tank structure and nominal normalised ground motion response spectral shape values at a structural damping of 7%
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(iii) Effective prestress available in ‘Class-A’ concrete in spherical tank portion of the structure between elevations 42.9 and 57.0 m.
The Bureau of Indian Standards (BIS) (IS 432, 1982; IS 456, 1978) specify the material strength requirements for concrete and reinforcing steel in terms of characteristic strength and S.D., the for-
Fig. 1. Structural features of the elevated water retaining structure.
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259
Fig. 2. Finite element model of the elevated water retaining structure.
mer being a reference value preferred for specifications and design. Characteristic strength is that value of the strength below which not more than 5% of the test results are expected to fall and it has a bearing on the uncertainties in establishing the intended strength of the structure. The material strength in actual structure may be somewhat
different than the characteristic strength, this being mostly due to the inherent variability of the material properties arising in the manufacture and errors or deviations arising in the construction. The deviations in concrete strength may arise due to errors in the mix proportions, presence of impurities, inadequate compaction or curing and
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construction faults. Therefore, considering the inherent variability of concrete strength during the production stage, it becomes necessary to design the concrete mix to have a target mean strength, which is higher than the characteristic strength. However, during the testing of samples for strength, some samples may have the strength which is less than the characteristic strength, these samples shall also be deemed to comply with the strength requirements if they fulfil the strength requirements stipulated in the BIS (IS 456, 1978). Considering these, the material strength for ‘Class-A’ concrete has been varied from 28 to 43 MPa and that for ‘Class-B’ concrete has been varied from 16 to 26 MPa. The deviations in reinforcing steel strength may arise largely due to the manufacturing defects such as the deviations from the nominal diameter and the metallurgical composition of the reinforcing steel bars. Sometimes inaccurate positioning of the reinforcing bars may result in deviation of anticipated strength of the individual members. However, BIS (IS 432, 1982), stipulates that, during the testing of samples for strength, no sample shall have the strength, which is less than the characteristic strength. Considering this, the material strength of the reinforcing steel has been varied from 250 to 325 Mpa by assuming a coefficient of variance of 10% for the material strength. The effective prestress in concrete, undergoes a gradual reduction with time from the stage of transfer due to various causes. This is generally referred to as loss of prestress which depends on several factors, such as the properties of the concrete and steel, method of curing, degree of prestress and process of prestressing. Therefore, it is difficult to generalise the exact amount of loss of prestress. However, based on the available data (Raju, 1990), typical values of the loss of prestress that could be encountered under normal conditions were considered and the effective prestress available in the ‘Class-A’ concrete in spherical tank portion of tank between elevations 42.9 and 57.0 m has been varied from 21 950 to 16 460 kN, which would be equivalent to 80– 60% of the initial prestressing.
3.3.2. Parameter for inelastic energy absorption An earthquake represents a source of limited energy, and therefore, it is normally expected that in the event of an earthquake occurrence, the structure must display reasonably good energy absorption characteristics by undergoing large deformations even beyond the yield without loosing much of its load carrying capacity and at the same time it should not fail to serve the various process features to which it is supposed to cater for. This ideal situation would require that the structure should be sufficiently ductile, so that it can deform inelastically beyond the yield and in doing so will redistribute the excess load to elastic parts of the structure. It may be noted that, in reinforced concrete structures, the ductility is largely governed by various factors viz. material properties of concrete and reinforcing steel, confinement of concrete, amount of reinforcing steel provided and the rate of loading and could be improved upon by proper structural detailing and control of construction procedures. Therefore, it may be mentioned that, the ductility available within the structure is greatly influenced by the variability associated with the construction materials (i.e. the concrete and the reinforcing steel) and the structural reinforcement detailing aspects. In the present case study, the only parameter considered under this category was ductility ratio available within the structure and in the absence of any available experimental data to establish the exact ductility ratio for the present structure, it has been varied from 1.0 to 1.75 to maintain reasonable conservatism in the seismic response analysis. No non-linear analyses were performed in the present study instead, the effect of deamplification resulting from inelastic dissipation was taken into account by using the ductility modified response spectra (Newmark and Hall, 1982; Chopra, 1998), which was derived from the elastic response spectra corresponding to median damping value and the ductility ratio. 3.3.3. Parameters for structural response For satisfactory performance in the event of an earthquake occurrence, generally the structures are analysed and designed by using specific deterministic response parameters for the structures,
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which may often be highly conservative. Since, many of these response parameters are random in nature with wide variability, most oftenly associated with them, the actual structural response may substantially differ from the structural response, calculated at the time of designing the structures. Therefore, the following variable parameters that may affect the calculated seismic response of the structures, were considered in the present case study under this category. (i) Spectral shape, representing the variability in ground motion and associated ground response spectra. (ii) Damping, representing the variability in response due to difference between actual damping and design damping. (iii) Soil–structure interaction, representing the variability in response due to variation in soil parameters for side soil and foundation soil. (iv) Fluid– structure interaction, representing the variability in response due to variation in the total height of water available inside the tank. For the aseismic design of structures as well as for the seismic re-qualification of the existing structures, the most important consideration is to derive a site-specific design basis response spectrum and at the same time the shape of the spectra is largely governed by the local geological and tectonic features. Therefore, the derivation of suitable design basis response spectral shape should be done by considering a large number of records having earthquake parameters in the range of interest and for better compatibility, these data should be drawn from sites having similar geological and tectonic conditions. The statistical combination of response spectra of number of individual records will then result in the site-specific design basis response spectra. However, it may be mentioned that, the mean (M) and standard deviation (S.D.) (|) of spectral values at all frequencies reflect the variations in the source-transmission-path site characteristics and help in assigning statistical confidence limits with specified spectral values to achieve an acceptable level of conservatism in the analyses. Generally, (M +|) response spectral shapes having a
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confidence level of 84% are used for aseismic design of structures for regions with low seismic risk. In the present case study, the site-specific design basis peak response spectral shape values corresponding to the time period of 0.2 s have been varied from 1.5395 to 3.8905, to represent the wider range of the variation in seismo-tectonic status of the region. For the vary purpose of seismic re-qualification of existing structures, it could be ideal to determine experimentally the important properties of the structures including its damping, but this is rarely done, generally, because of the unavailability of the experimental set-up at the actual structure site and due to lack of budget and time. Therefore, for an existing structure to be re-evaluated for its seismic safety, the selection of damping ratio is generally based on the available data on damping extracted from recorded earthquake motions of many structures of different types and different construction materials. However, in the present case study, due to unavailability of experimental data on damping, the damping ratios for the structure were taken from available data set (Newmark and Hall, 1982), and varied from 4 to 10%. It is widely recognised that the behaviour of a structure during an earthquake may be significantly affected by the nature of the foundation and surrounding soil media. Various authors have proposed various models of soil– structure interaction and simplified criteria to account for or ignore the soil– structure interaction. However, it has been observed that various criteria to ignore soil–structure interaction and carry out analysis with a fixed-base assumption depend on case specific parameters and as such the conservatism of fixed-base analysis can not be generalised (Patnaik et al., 1995; Ghosh et al., 1997). In the present case study, the uncertainties in soil–structure interaction are accounted for by considering the variation in shear wave velocity of soil and in the absence of adequate soil investigation data the shear wave velocity for side soil has been varied from 350 to 1400 m s − 1, while that for foundation soil has been varied from 1800 to 5000 m s − 1.
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One of the practical problems that generally occurs in the design of liquid-retaining structures, is the interaction between the contained liquid and the structure itself during the vibrations under the influence of seismic ground motion. The resulting horizontal hydrodynamic pressures, i.e. impulsive and sloshing due to fluid – structure interaction effects depend on the characteristics of the excitation, the properties and the level of the contained liquid and the geometric and physical properties of the liquid-retaining structure itself (USAEC, 1963). The fluid height largely affects the distribution of the impulsive and sloshing components of the hydrodynamic pressures along the height of the structure as well as the availability of effective fluid mass for the vertical seismic ground motion, and this, in turn, may affect the seismic structural response significantly. Consider-
ing these, the effective total water height in the present structure has been varied from 13.2 to 7.9 m in the present study. Table 2(a) summarises the range of variation of the relevant input data for the tank structure about their nominal values while the Table 2(b) gives the range of variation of the normalised ground motion response spectral shape values about their nominal values.
3.4. Failure mode for the present case study The first step in developing a fragility description, is to identify the failure mode associated with the fragility curves and the consequences of the failure mode. Identification of the most likely failure modes, caused by the seismic event, is generally governed by the experience and judge-
Table 2 Range of variation Category
Parameter
Rele6ant input data for the present tank structure (a) (a) Material strength (i) ‘Class-A’ concrete (ii) ‘Class-B’ concrete (iii) Reinforcing steel (iv) Effective prestress in the spherical tank portion between elevation 42.9 and 57.0 m (b) Inelastic energy (i) Ductility ratio available with in the absorption structure (c) Structural response (i) Structural damping (ii) Shear wave velocity for the side soil (iii) Shear wave velocity for the bottom foundation soil (iv) Total water height in the tank Normalised ground motion response spectral shape 6alues at a structural damping of 7% (b) Time period (s) RSSV 0.0 1.000 (constant) 0.1 1.4055–3.1680 0.2 1.5395–3.8905 0.4 1.1545–3.2255 0.6 0.7805–2.4595 0.8 0.5365–1.9235 1.0 0.4280–1.7620 1.5 0.2380–1.4820 2.0 0.1355–1.2215 Category: structural response; parameter: response spectral shape.
Range of variation
28–43 MPa 16–26 MPa 250–325 MPa 21 950–16 460 KN 1.0–1.75 4–10% 350–1400 m s−1. 1800–5000 m s−1. 13.2–7.9 m
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ment of the structural analyst engaged in generating the fragility curves. In general, the structures are considered to fail functionally when the degree of distress in the structures due to seismic event achieves a certain level beyond which, it would be reasonable to assume that the appropriate and designated functioning of the safety-related equipment attached to the structures, could be interfered substantially. However, this degree of distress is very judgmental and must reflect the location of the structural distress and the attached safety-related equipment. It may be further noted that the failure is characterised by a cumulative distribution function which describes the probability that failure has occurred, given a value of loading and in the context of seismic fragility this loading could be represented in terms of PGA, depending on the structure and the failure mode under consideration. In the present case study, it would be reasonable to consider that, under seismic excitation, the present elevated water-retaining structure essentially behaves like a cantilever beam with circular cross-section, somewhat modified by the rocking of the base raft. Therefore, it would be appropriate to assume bending of structure in the direction of seismic ground motion as a dominant mode of failure and this will result in the linear variation of the axial bending stresses across the cross-section as long as the response of the structure stays in the elastic range. The structure was considered to fail when the axial bending stress due to seismic event achieves the permissible axial bending stress at the locations of interest. For ‘Class-A’ concrete, the compressive axial bending stress due to seismic event shall be resisted by the concrete and reinforcing steel together, while the tensile axial bending stress shall be resisted by the prestressing tendons and the reinforcing steel together. It may be noted that, for ‘Class-A’ concrete, only after the tensile axial bending stress due to seismic event exceeds the available prestress, will the bonded reinforcing steel experience the load. For ‘Class-B’ concrete, the compressive axial bending stress shall be resisted by the concrete and reinforcing steel together, while the tensile
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axial bending stress shall be resisted by the reinforcing steel alone. The typical axial bending stress distribution across the cross-section due to seismic ground motion in one of the direction is shown in Fig. 3. Table 3 gives the typical values of permissible axial bending stresses at different locations considered in the present case study, for the nominal values of the input data. Here, it may be mentioned that, while calculating the permissible axial bending stresses at different locations the non-seismic portion of the total load acting on the structure has been subtracted from the available capacities, therefore, the values mentioned in the Table 3 represent the portion of axial bending stress available for the seismic load only.
3.5. Numerical analyses The present elevated water-retaining structure has been analysed by using the response spectrum technique. In the response spectrum method of analysis, linear elastic structures subjected to base excitation are analysed by solving the uncoupled equations of motion for maximum modal responses. The total response of the structure is obtained by summing up the contribution of response due to each mode. In the present case study, the seismic input includes two horizontal and the vertical components of the ground motions in combination. For the present structure the maximum response has been obtained by using CQC method for modal combination (ASCE, 1995) and the spatial combination for horizontal and vertical ground motions has been performed using the SRSS technique (ASCE, 1995). Software program COSMOS/M (COSMOS/M, 1997) was used for all analyses purposes. The analyses of the present tank structure were carried out for two conditions: (i) tank empty and (ii) tank filled with water. The PGA to failure was determined at two different locations: (i) node ‘9’ for the element ‘8’ and (ii) node ‘20’ for the element ‘19’ as shown in the Fig. 2 and these two elements were chosen, because of the maximum stress experienced by them due to given ground motion.
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Fig. 3. Typical axial bending stress distribution across the cross-section of wall.
Parametric studies were made by varying the material strengths of ‘Class-A’ concrete, ‘ClassB’ concrete and reinforcing steel, effective prestress available in ‘Class-A’ concrete in spherical tank portion, ductility ratio and damping available within the structure, normalised ground motion response spectral shapes, shear wave ve-
locity for the surrounding and foundation soil media and the total height of water in the tank about their nominal values making change only in one variable parameter at a time. It may be noted here that, variations in ductility ratio available within the structure and the total water height in the tank were considered only for
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the tank filled with water condition for the present case study. The results of the parametric studies were then used to determine the seismic safety factor (i.e. the ratio of PGA for failure to the design basis PGA) and its variability (i.e. its logarithmic mean value and logarithmic S.D.) with each of the parameters mentioned above for the bending mode of failure. Here, it may be mentioned that while calculating the variability for each of the parameters, different weights were assigned to the range of variation for the parameter considered at a time as per the probability of its occurrence, with the maximum weight assigned to the nominal value of the parameter which is the median value of the parameter. Here, while assigning the weights it was further assumed that, the variable parameters follow log-normal probability distribution function. In the absence of sufficient experimental data the logarithmic S.D. due to randomness about the median value (iR) due to individual parameters have been evaluated through a parametric study and later on the logarithmic S.D. due to uncertainties in the median values (iU) were evaluated based on the range of the iR and iU available in the literature (Kennedy and Ravindra, 1984) for the present case study. After the evaluation of variabilities due to individual parameters, they were combined to determine the overall logarithmic S.D. of the seismic safety factor due to randomness about the median value and due to the uncertainties in the median values for the structure at two different locations as mentioned above. Here it may be noted that the overall logarithmic S.D.
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of the seismic safety factor for the structure is the SRSS combination of the logarithmic S.D. of the individual parameters considered in the analyses. For the present case study Table 4(a and b), Table 5(a and b) give the calculated values of iR and iU at two different locations of the present structure (i.e. at node ‘9’ of element ‘8’ and node ‘20’ of element ‘19’). Table 6 summarises the values of median ground acceleration capacity (A( ) and composite variability (iC) at the locations of interest for the different tank conditions. In the present case study the seismic fragility of the structure in question has been evaluated considering the variation in the number of parameters as mentioned previously. Figs. 4–11 represent the seismic fragility of the present structure developed as families of conditional probability curves plotted against PGA at two different locations of interest for the different tank conditions.
4. Discussion of results and conclusions The seismic fragility curves are generated in the present case study considering the log-normal distribution of the various parameters which may be justified as an appropriate distribution, because of the fact that the statistical variation of many material properties and seismic structural response variables may be represented reasonably by this distribution. Moreover, by using the log-normal distribution for the variable parameters, the entire fragility curve and its uncertainty can be expressed by only three
Table 3 Typical values of permissible axial bending stress for the nominal values of the relevant input data Location
Node ‘9’ of element ‘8’ Node ‘20’ of element ‘19’
Tank condition
(i) Tank empty (ii) Tank filled with water (i) Tank empty (ii) Tank filled with water
Permissible axial bending stress (MPa) Compression
Tension
7.7362 4.3319 8.1658 7.0484
1.7305 5.1348 1.4438 2.5612
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Table 4 Calculated values of iR and iU for location at node ‘9’ of element ‘8’ Category
Parameter
(a) Tank empty condition (a) Material strength (i) ‘Class-A’ concrete (ii) ‘Class-B’ concrete (iii) Reinforcing steel (b) Structural response (i) Structural damping (ii) Response spectral shape (iii) Shear wave velocity for the side soil (iv) Shear wave velocity for the bottom foundation soil Overall variability (b) Tank filled with water condition (a) Material strength (i) ‘Class-A’ concrete (ii) ‘Class-B’ concrete (iii) Reinforcing steel (b) Inelastic energy (i) Ductility ratio available with in the structure absorption (c) Structural response (i) Structural damping (ii) Response spectral shape (iii) Shear wave velocity for the side soil (iv) Shear wave velocity for the bottom foundation soil (v) Total water height in the tank Overall variability
parameters namely median ground acceleration capacity (A( ), logarithmic S.D. due to randomness about the median value (iR) and logarithmic S.D. due to uncertainty in the median value (iU), which are easier to estimate. From Table 4(a and b), Table 5(a and b), it is very clear that, a substantially greater amount of variability (both iR and iU) is, because of the contribution from material strengths for ‘Class-A’ and ‘Class-B’ concrete, normalised ground motion response spectral shapes, ductility ratio available with in the structure and total water height available in the tank, therefore, the response of the present structure at the locations of interest appears to be quite sensitive to the variation of these parameters in the present case study. Similarly, it has also been observed from the same tables that, the overall variability due to uncertainty in median value (iU) is more than the overall variability due to randomness about the median value (iR) for bending compression mode of failure while the overall iU is
Bending compression
Bending tension
iR
iU
iR
iU
0.0003 0.1698 0.0054 0.0570 0.2205 0.0271 0.0285 0.2869
0.0005 0.2971 0.0094 0.0570 0.1103 0.1220 0.1284 0.3676
0.0003 0.0156 0.0300 0.0570 0.2279 0.0273 0.0285 0.2406
0.0005 0.0273 0.0525 0.0570 0.1140 0.1228 0.1284 0.2265
0.0024 0.2941 0.0095 0.0672
0.0042 0.5147 0.0167 0.1378
0.0025 0.0162 0.0104 0.0685
0.0043 0.0284 0.0183 0.1405
0.0617 0.2840 0.0349 0.0245 0.2114 0.4712
0.0617 0.1420 0.1572 0.1104 0.2114 0.6243
0.0616 0.2840 0.0350 0.0247 0.0701 0.3103
0.0616 0.1240 0.1575 0.1110 0.0701 0.2948
observed to be less than the overall iR for the bending tension mode of failure. This may be attributed to the fact that the evaluation of axial bending tension capacities are mainly governed by the contribution from reinforcing steel only and are supposed to be free from the uncertainty associated with the material strength parameters for concrete. From Table 6, it can reasonably be inferred that, in the present case study, lesser composite variability is associated with the tank empty condition as compared with the tank filled with water condition which reflects the significant contribution from the total water height towards the composite variability for the seismic structural response. Figs. 4–11 represent the seismic fragility of the structure at the locations of interest for different modes of bending failure, different tank conditions and different levels of confidence. In the present case study, the overall logarithmic S.D. due to randomness about the median value (iR) at different locations and for differ-
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Table 5 Calculated values of iR and iU for location at node ‘20’ of element ‘19’ Category
Parameter
(a) Tank empty condition (a) Material (i) ‘Class-A’ concrete strength (ii) ‘Class-B’ concrete (iii) Reinforcing steel (iv) Effective prestress in the spherical tank portion between elevations 42.9 and 57.0 m (b) Structural (i) Structural damping response (ii) Response spectral shape (iii) Shear wave velocity for the side soil (iv) Shear wave velocity for the bottom foundation soil Overall variability
(b) Tank filled with water condition (a) Material (i) ‘Class-A’ concrete strength (ii) ‘Class-B’ concrete (iii) reinforcing steel (iv) Effective prestress in the spherical tank portion between elevations 42.9 and 57.0 m (b) Inelastic (i) Ductility ratio available with in the structure energy absorption (c) Structural (i) Structural damping response (ii) Response spectral shape (iii) Shear wave velocity for the side soil (iv) Shear wave velocity for the bottom foundation soil (v) Total water height in the tank Overall variability
Bending compression
Bending tension
PR
PU
PR
PU
0.1490
0.2607
0.0076
0.0133
0.0070 0.0031 0.0029
0.0122 0.0055 0.0051
0.0070 0.0187 0.0179
0.0122 0.0328 0.0313
0.0605
0.0605
0.0606
0.0606
0.2445 0.0130 0.0020 0.2930
0.1222 0.0583 0.0091 0.3004
0.2444 0.0130 0.0021 0.2537
0.1222 0.0580 0.0094 0.1563
0.1675
0.2930
0.0030
0.0052
0.0046 0.0036 0.0033
0.0080 0.0063 0.0059
0.0045 0.0106 0.0098
0.0078 0.0185 0.0171
0.1190
0.2440
0.1190
0.2439
0.0609
0.0609
0.0609
0.0609
0.3030 0.0049 0.0009 0.1765 0.4111
0.1515 0.0221 0.0040 0.1765 0.4515
0.3030 0.0049 0.0010 0.1167 0.3515
0.1515 0.0221 0.0045 0.1167 0.3178
Table 6 Median ground acceleration capacity (A( ) and composite variability (iC) at different locations of interest for the present tank structure Location
Node ‘9’ of element ‘8’ Node ‘20’ of element ‘19’
Tank condition
Tank Tank Tank Tank
empty filled with water empty filled with water
Bending compression
Bending tension
A( (g)
iC
A( (g)
iC
3.4391 0.3102 1.5406 0.3885
0.4663 0.7822 0.4196 0.6106
0.7693 0.3676 0.2724 0.1412
0.3305 0.4280 0.2980 0.4739
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Fig. 4. Seismic fragility curves for location at node ‘9’ of element ‘8’.
Fig. 5. Seismic fragility curves for location at node ‘9’ of element ‘8’.
ent tank conditions, was evaluated through parametric studies and later on the values for overall logarithmic S.D. due to uncertainty in the median value (iU) are evaluated based on the data available in the literature (Kennedy and Ravindra, 1984).
However, it may be mentioned that, the factors iR and iU should be evaluated separately for the structure, and this, in turn, requires performing a lot of experiments and analyses in order to find the variations of different variable parameters.
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In the present case study, the seismic response of the structure at the locations of interest has been found to be quite sensitive to the randomness
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and uncertainty associated with the material strength of concrete, normalised ground motion response spectral shapes, ductility ratio variable
Fig. 6. Seismic fragility curves for location at node ‘9’ of element ‘8’.
Fig. 7. Seismic fragility curves for location at node ‘9’ of element ‘8’.
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Fig. 8. Seismic fragility curves for location at node ‘20’ for element ‘19’.
Fig. 9. Seismic fragility curves for location at node ‘20’ of element ‘19’.
available within the structure and the total water height in the tank. By carrying out extensive experiments combined with the analysis work to ascertain the variation in the various variable
parameters, which may contribute towards the randomness, and uncertainty in the seismic structural response, the fragility evaluation may be carried out in a more realistic manner.
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Fig. 10. Seismic fragility curves for location at node ‘20’ of element ‘19’.
Fig. 11. Seismic fragility curves for location at node ‘20’ of element ‘19’.
Appendix A. Nomenclature a A A(
any value of ground acceleration peak ground acceleration, ground acceleration capacity, a random variable median ground acceleration capacity of a structure
Adesign ‘Class-A’ ‘Class-B’ DS
design basis PGA value prestressed concrete (nominal value of FCK = 35 MPa) reinforced concrete (nominal value of FCK = 20 MPa) random variable accounting for the change in input motion at foundation depth
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F F( ) FCK FY M MC
P Pf P %f
Q RS RSSV S SC
SD
SRSS SS
SSI iC iF iR iU
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factor of safety above design basis PGA factor of safety for the variable specified in the subscript characteristic strength for concrete characteristic yield strength for steel random variable reflecting the variability due to modelling assumptions random variable reflecting the variability due to method of combination of various modes of vibrations exceedence probability level conditional frequency of failure at any exceedence probability level (includes both randomness and uncertainty) conditional frequency of failure at any non-exceedence probability level (includes both randomness and uncertainty) non-exceedence probability level structural response response spectral shape values strength of a structure random variable accounting for the variability due to method of earthquake component combination random variable reflecting the variability due to difference in actual and design structural damping square root of sum of squares random variable reflecting the variability in the response spectral shapes used in the design random variable reflecting the variability due to the effects of soil-structure interaction composite logarithmic S.D. logarithmic S.D. of ‘F’ logarithmic S.D. representing the inherent randomness in the variable logarithmic S.D. representing the uncertainty in the variable
v ( )
ductility ratio standard Gaussian cumulative distribution function −1( ) inverse of standard Gaussian cumulative distribution function
References ASCE Standards, 1995. Seismic Analysis of Safety-Related Nuclear Structures and Commentary, Fourth Revision, American Society of Civil Engineers, Washington, pp. 3.18 – 3.28. Chopra, A.K., 1998. Dynamics of Structures, Theory and Applications to Earthquake Engineering. Prentice-Hall, New Delhi, India, pp. 269 – 275. COSMOS/M 2.0, 1997. A Complete Finite Element Analysis System, Structural Research & Analysis Corporation, Los Angeles. Ghosh, A.K., Agrawal, M.K., Patnaik, R., Bhargava, K., 1997. Effect of soil parameters on the seismic response of a 3-D frame structure. Transaction of the 14th International Conference on Structural Mechanics in Reactor Technology, Lyon, France, pp. 331 – 338. IS: 432 (Part-I), 1982. Indian Standard Specification for Mild Steel and Medium Tensile Steel Bars and Hard-Drawn Steel Wire for Concrete Reinforcement, Third revision, Bureau of Indian Standards, New Delhi, pp. 3 – 9. IS: 456, 1978. Indian Standard Code of Practice for Plain and Reinforced concrete, Third Revision, Bureau of Indian Standards, New Delhi, pp. 40 – 45. Kennedy, R.P., Ravindra, M.K., 1984. Seismic fragilities for nuclear power plant risk studies. Nucl. Eng. Des. 79, 47 – 68. Kennedy, R.P., Cornell, C.A., Campbell, R.D., Kaplan, S., Perla, H.F., 1980. Probabilistic seismic safety study of an existing nuclear power plant. Nucl. Eng. Des. 59, 315 –338. Newmark, N.M., Hall, W.J., 1982. Earthquake Spectra and Design. Earthquake Engineering Research Institute, Berkeley, CA, pp. 29 – 54. Patnaik, R., Bhargava, K., Ghosh, A.K., Agrawal, M.K., 1995. Effect of foundation conditions on the seismic response of a space frame. Transactions of the 13th International Conference on Structural Mechanics in Reactor Technology, Porto Alegre, Brazil, pp. 163 – 168. Raju, N.K., 1990. Prestressed Concrete, second ed. Tata McGraw-Hill, New Delhi, pp. 73 – 79. USAEC, 1963. Nuclear Reactors and Earthquakes (TID7024), US Atomic Energy Commission, Washington, pp. 185 – 188.