Architectural Engineering November 2013, Volume 1, Issue 3, PP.52-59
Earthquake Loss Evaluation of Buildings Based on Story EDP-DV Functions Zhaoping Jia†, Ziyan Wu, Qi’ang Wang School of Civil Engineering, Northwestern Polytechnical University, Xi’an 710129, China †Email:
Abstract In order to quantify the seismic performance of buildings reasonably and effectively, the paper puts forward the method of earthquake loss evaluation of buildings based on story EDP-DV functions. Firstly, considering the randomness of seismic response parameters, the distribution functions of story EDPs at a given level of ground motion intensity can be achieved through Incremental Dynamic Analysis. Meanwhile, story EDP-DV functions which relate story response parameters (story EDPs) directly to economic losses (DVs) can be created beforehand by integrating component fragility functions and loss functions; they can be called directly without double computation for the same type of buildings. Lastly, the economic losses of a building at a given level of ground motion intensity can be achieved by combining the distribution functions of story EDPs and story EDP-DV functions. On this occasion, the proposed method omits the damage measure in PEER methodology and makes the story not component as a unit of account to evaluate the earthquake losses of a building reasonably and accurately. The numerical example shows that the proposed method is feasible and reasonable to quantify the seismic performance of buildings. Keywords: Seismic Performance; Earthquake Loss Evaluation; Story EDP-DV Functions; Distribution Functions; Incremental Dynamic Analysis
1 INTRODUCTION Despite current seismic codes can protect life-safety very well by providing a set of prescriptive provisions, they can’t quantify the seismic performance of buildings explicitly. One way of quantifying the seismic performance that has been proposed by recent researches [1-3] is using economic losses as a metric to gauge how well structural systems respond when subjected to seismic ground motions. The uncertainties of ground motion (i.e. earthquake source, earthquake magnitude…etc) and structure (i.e. strength of materials, geometry size, structural style…etc) led to the variation of earthquake losses, so it’s important for performance-based seismic design to evaluate earthquake losses of buildings reasonably and accurately. Vision 2000 defined the performance levels and multiple performance targets clearly, but the performance levels defined in the document were often qualitative, not well-defined and, consequently, open to subjectivity. Ma and Lv regarded the discrete performance of structures as a continuous variable to estimate the economic losses of a building, but the method only considered the losses under basic earthquake. Porter and Kiremdjian applied the Monte Carlo simulation to develop fragility functions for common building assemblies. In this case, it could predict building-specific relationships between expected losses and seismic intensity. The calculation was very huge. Rahnama and Wang  applied the fragility functions to quantify the average loss ratio of similar buildings at a given level of ground motion intensity, but the paper focused on the estimation of average annual losses. The deficiencies of above researches were that the model was too simple or complex, so they couldn’t quantify the seismic performance of buildings effectively. The Pacific Earthquake Engineering Research (PEER) Center has established a fully probabilistic framework that used the results from seismic hazard analysis and response simulation to estimate damage and monetary losses incurred during earthquakes. The methodology was divided into four basic stages that account for the following: ground motion hazard estimation, response estimation, damage estimation, loss estimation. The above four stages corresponded to the variables respectively: intensity measure (IM, - 52 http://www.ivypub.org/AE/
i.e. peak ground acceleration, spectral acceleration…etc), engineering demand parameter (EDP, i.e. interstory drift ratio, peak floor acceleration…etc), damage measure (DM), decision variable (DV, i.e. economic loss, loss of life, downtime…etc). The results of each stage serve as input to the next stage as shown in schematically in Fig.1. IM
FIG.1 PEER METHODOLOGY
The previous PEER methodology involves several integrations of many random variables making it very computational intensive. Based on the methodology, the paper puts forward the method of earthquake loss evaluation of buildings based on story EDP-DV functions. The proposed method combines the distribution functions of story EDPs and story EDP-DV functions to compute the economic losses of a building. The story EDP-DV functions which relate story EDPs directly to DVs can be created beforehand by applying empirical data (component fragility functions and loss functions), and they can be called directly without double computation for the same type of buildings. So the proposed method omits the damage measure (DM) in PEER methodology and promotes the component-based earthquake loss evaluation to story-based earthquake loss evaluation, which can quantify the seismic performance of buildings reasonably and effectively.
2 DISTRIBUTION FUNCTIONS OF STORY EDPS 2.1 Incremental Dynamic Analysis (IDA) The IDA curves indicate the relationship between IM and EDP. In the process of establishing IDA curves, the IM is monotonically increasing and the EDP should be able to express the structural performance properly. The IDA method firstly selects some seismic waves for the site being considered, and each seismic wave will be magnified to different levels of seismic intensity proportionally. The EDP can be received by the nonlinear time-history analysis of a structure at a given level of ground motion intensity for each seismic wave. The nonlinear time-history analysis doesn’t end until the structure collapses at a high level of ground motion intensity. Corresponding to a seismic wave, an IDA curve can be received by spline interpolation of each (EDP, IM) point. The method can fully grasp the entire process of the structure which is from the elastic analysis to the elastic-plastic analysis.
2.2 Distribution Functions of Story EDPs at a Given Level of Ground Motion Intensity The uncertainty of earthquake action is much greater than material strength, load and so on, so the paper only considers the variation of seismic response parameters. Shome and Aslani assume that the structural response parameters at the intensity level IM are lognormally distributed when the structure is non-collapse. According to the IDA curves, the distribution functions of story EDPs, when the building collapse has not occurred at the intensity level IM is given as follows: Ln( x) Ln ( EDP ) (1) F x NC , IM im = P （EDPk x NC , IM im) = Ln( EDP ) where x is a random variable of story EDPs, Ln ( EDP ) and Ln ( EDP ) are logarithmic mean and logarithmic standard deviation of story EDPs, respectively, given that the building collapse has not occurred and the level of ground motion intensity is IM.
Equation (1) can’t express the distribution functions of story EDPs when the building collapses (the functions are equal to 0 actually). The distribution functions of story EDPs conditioned on ground motion intensity, F ( x IM im) , can be computed by conditioning it on mutually exclusive and collectively exhaustive events that can be experienced by the building, namely collapse (C) and non-collapse (NC), of the building ,as follows: Lnx Ln ( EDP ) F x IM im = P （EDPk x IM im) = 0 P C IM im + P NC IM im Ln ( EDP ) (2) Lnx Ln ( EDP ) 1 - P C IM im Ln ( EDP )
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P(C IM im) u / n
where F x IM im is the distribution function of story EDPs conditioned on IM, P( NC IM im) is the probability that the structure will not collapse conditioned on IM, P(C IM im) is the probability that the structure will collapse conditioned on IM, which is complementary to P( NC IM im) , u and n are the number of seismic waves due to collapse and total seismic waves, respectively, conditioned on IM.
3 STORY EDP-DV FUNCTIONS 3.1Component Fragility Functions and Loss Functions from Empirical Data Component fragility functions provide the probability of experiencing or exceeding a particular damage state conditioned on component EDP. It has been well-established that the lognormal distribution provides relatively good fit to empirical cumulative distributions computed from experimental data [2, 10]. The paper takes the logarithmic mean and logarithmic standard deviation of component fragility functions [2, 10] to compute the component fragility （DS dsi EDPj edp j ）. functions, P Component loss functions are the component’s economic losses conditioned on component damage state. In order to reduce computation, the paper introduces the normalized loss which is equal to the loss cost divided by replacement cost. The component’s normalized losses conditioned on component damage state (component loss function), E '[ L j DS dsi ] , can be achieved from[2,10].
3.2 Story EDP-DV Functions Story EDP-DV functions provide the normalized story losses conditioned on story EDP. The story EDP-DV functions which relate story EDPs directly to DVs can be created beforehand by integrating component fragility functions and loss functions, and they can be called directly without double computation for the same type of buildings. So the proposed method omits the damage measure (DM) in PEER methodology and promotes the component-based earthquake loss evaluation to story-based earthquake loss evaluation. The first step in developing story EDP-DV functions is collapsing out the third intermediate step of DM in PEER methodology by combining information from component fragility functions and loss functions as shown in equation(4). l
E ' [ L j EDPj edp j ] E ' [ L j DS dsi ]P( DS dsi EDPj edp j )
1 - P（DS dsi+1 EDPj = edp j ） i0 P（DS = dsi EDPj = edp j ）= P（DS dsi EDPj = edp j ）- P（DS dsi+1 EDPj = edp j ） 1 i l (5) il P（DS dsi EDPj = edp ） j where E ' [ L j EDPj edp j ] is the normalized loss in the j th component when it is subjected to a component EDP, E ' [ L j DS dsi ] is the normalized loss in the j th component when it is in damage state i (component loss function), and P( DS dsi EDPj edp ) is the probability of the j th component being in damage state i given that it is j subjected to a component EDP. The probability of being in each damage state for component j can be obtained from component fragility functions as shown in equation(5).
The second step involves summing the individual component losses for the entire story of a building. Previously, this summation requires inventorying the number of components and the value of each component type. However, the paper assumes that components of the same type are grouped together in the same story and experience the same level of damage. The loss for each component type can be calculated by multiplying the results of equation (4) by its value relative to entire value of the story. Component types can then be summed for the entire story, as follows: s
E ' [ LSTORY EDPk edpk ] b j E ' [ L j EDPj edp j ] j 1
where E ' [ LSTORY EDPk edpk ] is the normalized loss of the k th story conditioned on the story EDP, and this is how the story EDP-DV functions will be expressed. b j is equal to the total value of components of the same type in - 54 http://www.ivypub.org/AE/
the k th story divided by the total value of the k th story, termed “component cost distribution”, which can be achieved from 2007 RS Means Square Foot Costs .
4 EARTHQUAKE LOSS EVALUATION The normalized story losses conditioned on IM can be computed using the total probability theorem as follows:
E '[ LSTORY IM im] E ' [ LSTORY x] dP( EDPk x IM im) E ' [ LSTORY x] d (1 P( EDPk x IM im))
E [ LSTORY x] d (1 F ( x IM im))
where E '[ LSTORY IM im] is the normalized story loss conditioned on IM, E ' [ LSTORY x] is the story EDP-DV function, F ( x IM im) is the distribution function of story EDPs conditioned on IM. The normalized building losses are computed as the sum of the losses in each story of the building conditioned on IM as follows: q
E '[ L IM im] E '[ LSTORY IM im]ck
where E '[ L IM im] is the normalized building loss conditioned on IM, q is the number of stories in the building, ck is equal to the total value of k th story divided by the total value of the building. In order to figure out the direction of controlling losses, the normalized building losses conditioned on IM can be disaggregated into mean losses due to collapse, and mean losses due to non-collapse as follows: ' (9) E '[ L IM ] EC' ’[ L IM ] ENC [ L IM ] The normalized building losses are equal to 1 when the building collapses, so EC' ’[ L IM ]=P(C IM ) E '[ L C ]=P(C IM ) ' ENC [ L IM ]=E '[ L IM ] EC' ’[ L IM ]
' where EC' ’[ L IM ] and ENC [ L IM ] are mean losses due to collapse and mean losses due to non-collapse respectively, E '[ L C ] is the normalized building loss when the building collapses, P(C IM ) is the probability that the structure will collapse conditioned on IM and can be achieved from equation (3).
5 CASE STUDY The paper takes a six layer reinforced concrete frame as a research object. The number of longitudinal spans and transversal spans are 8 and 3 respectively, and each span length is 4m; each interstory height is 3m. Columns and beams are 500mm×500mm and 300mm×500mm respectively, and the sickness of each plate is 100mm. The site being considered is classified as Ⅱ, and the stories are divided into first floor, typical floor and top floor in the building. A three-dimensional analytical model of the building is generated using the finite element platform SAP2000 as shown in Fig.2. The material of concrete and steel are C30 and HRB335 (elasticity modulus, EC=30Gpa and ES=200Gpa; Poisson’s ratio, μc=0.2 and μs=0.3), and the density of reinforced concrete is 2500kg/m3. The columns and beams adopt M3 and P-M2-M3 respectively for reinforcement design, and the plates adopt Shell-Thin element. Plastic hinges are most likely to occur at each end of the columns and beams, and they are modeled using Plastic (Wen) elements.
FIG.2 TRI-DIMENSIONAL FRAME MODE - 55 http://www.ivypub.org/AE/
The site being considered is classified as â…Ą, so 20 seismic waves which fit the classified site are selected from the database of PEER. Here the parameter used to represent IM at the site is the spectral acceleration of a linear elastic 5% damped single-degree-freedom system with a period of vibration equal to the fundamental period of vibration of the building, Sa. Fig.3 shows the acceleration response spectrum of 20 seismic waves.
FIG.3 ACCELERATION RESPONSE SPECTRUM
FIG.4 IDA CURVES OF TYPICAL FLOOR
5.1 Distribution Functions of Story EDPs Because structural components and some nonstructural components are drift-sensitive, and the other nonstructural components are acceleration- sensitive, the EDPs chosen in this study are interstory drift ratio (IDR) and peak floor accelerations (PFA). According to the above IDA method, the spectral acceleration of each seismic wave ,Sa , will be magnified to 0.05g,0.15g,0.25g,0.35g,0.45g,0.55g,0.65g,0.75g,0.85g,0.95g,1.05g and 1.20g. In this case, the seismic waves of amplitude modulation will be inputted into the finite element model to establish the IDA curves. Corresponding to 20 seismic waves, the paper only lists 20 IDA curves of the typical floor with EDP choosing IDR as shown in Fig.4. According to the IDA curves of each story in the building, the distribution functions of story EDPs conditioned on IM are established. Fig.5(a) and Fig.5(b) only show the distribution functions of IDR and the distribution functions of PFA conditioned on Sa=0.05g at each story. 1 0.9
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 x=IDR
First Floor Typical Floor Top Floor
First Floor Typical Floor Top Floor
FIG.5 DISTRIBUTION FUNCTIONS OF STORY EDPS CONDITIONED ON Sa=0.05g: (a) IDR AT EACH STORY, (b) PFA AT EACH STORY - 56 http://www.ivypub.org/AE/
5.2 The Establishment of Story EDP-DV Functions According to the empirical data (component fragility functions and loss functions) [2,10] and the component cost distribution as shown in table 1, the paper establishes the story EDP-DV functions at each story for mid-rise, ductile reinforced concrete frame buildings as shown in Fig.6. TABLE 1 COMPONENT FRAGILITY FUNCTION & LOSS FUNCTION & COST DISTRIBUTION IN MID-RISE BUILDINGS WITH DUCTILE REINFORCED CONCRETE FRAMES Component
Fragility Function Parameters Median Dispersion
DS1 DS2 DS3 DS4 DS1 DS2 DS3 DS1 DS2 DS3 DS1 DS1 DS2 DS3 DS1 DS2 DS3 DS4 DS1 DS2 DS3 DS1 DS2 DS3 DS4
0.0070 0.0170 0.0390 0.0600 0.0040 0.0100 0.0900 0.0021 0.0069 0.0127 0.0127 0.0160 0.0320 0.0360 0.0055 0.0100 0.0220 0.0350 0.30g 0.65g 1.28g 0.70g 1.00g 2.20g 3.50g
0.45 0.50 0.30 0.22 0.39 0.25 0.24 0.61 0.40 0.45 0.45 0.29 0.29 0.27 0.60 0.50 0.40 0.35 0.40 0.50 0.55 0.50 0.50 0.40 0.35
0.14 0.47 0.71 2.25 0.10 0.40 2.75 0.10 0.60 1.20 1.20 0.10 0.60 1.20 0.03 0.10 0.60 1.20 0.12 0.36 1.20 0.02 0.12 0.36 1.20
0.5 0.4 0.3
0.2 First Floor Typical Floor Top Floor
First Floor Typical Floor Top Floor
FIG.6 STORY EDP-DV FUNCTIONS IN MID-RISE BUILDINGS WITH DUCTILE REINFORCED CONCRETE FRAMES
Comparing plots in Fig.6(a) between the different floor types shows that losses for drift-sensitive components are higher for the 1st and typical floors than the top floor. Conversely, Fig.6(b) shows that acceleration-sensitive components appear the opposite trend, where the larger losses are observed in the top floor. The reason is that drift-sensitive items, such as structural components, make more of the relative story value at the first floor. On the - 57 http://www.ivypub.org/AE/
other hand, acceleration-sensitive components may make up more of the story value at the top floor, because acceleration-sensitive items, such as ceilings, are typically located on the roof of these types of buildings.
5.3 Earthquake Loss Evaluation The distribution functions of story EDPs conditioned on IM and story EDP-DV functions are substituted into equation (7) for getting the normalized story losses conditioned on IM. In this case, the normalized story losses are turned into the normalized building losses conditioned on IM by equation(8). In order to figure out the direction of controlling losses, the normalized building losses conditioned on IM are disaggregated into mean losses due to collapse and mean losses due to non-collapse by equation(9) and (10)as shown in Fig.7. 1 Total Collapse Non-Collapse
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
FIG.7 THE BUILDING LOSSES CONDITIONED ON IM (WITH LOSS DISAGGREGATION)
As shown in Fig.7, the normalized building losses increase fairly linearly with increasing levels of ground motion intensity. According to the current China code for seismic design of buildings , the normalized building losses are 10.37% under frequent earthquake (Sa=0.16g), 34.98% under basic earthquake (Sa=0.45g) and 73.82% under rare earthquake (Sa=0.90g). The calculation above can fairly meet the requirements of no damage under frequent earthquake, repairable damages under basic earthquake and no collapse under rare earthquake, so the method of earthquake loss evaluation of buildings based on story EDP-DV functions is reasonable. The building losses are small under frequent and basic earthquake, so the building can be used again by the simple repair. Conversely, the building losses are large under rare earthquake, so the building exists so serious hidden danger that the repair is typically not cost effective. The results of loss evaluation can help stakeholders make more informed design decisions and quantify the buildingâ€™s seismic performance reasonably and effectively. According to the loss disaggregation in Fig.7, the building losses are primarily dominated by the non-collapse losses under frequent and basic earthquake. For the building, non-collapse losses reach the peak, 0.41, at an approximate Sa of 0.63g and collapse losses do not begin to dominate until an intensity of 0.81g. It figures out the direction of controlling losses conditioned on different levels of ground motion intensity.
6 CONCLUSIONS (1)The method omits the DM in PEER methodology and promotes the component-based loss evaluation to story-based loss evaluation by story EDP-DV functions, so it can quantify the buildingâ€™s seismic performance reasonably and effectively. (2)The paper considers not only the drift-sensitive component losses but also the acceleration-sensitive ones, so the loss evaluation results of the proposed method are conservative. (3)The building losses are small under frequent and basic earthquake, so the building can be used again by the - 58 http://www.ivypub.org/AE/
simple repair. Conversely, the building losses are large under rare earthquake, so the building exists so serious hidden danger that the repair is typically not cost effective. (4)The building losses are mainly caused by the structure non-collapse under frequent and basic earthquake. Conversely, the building losses are basically caused by the structure collapse under rare earthquake. In this case, the direction of controlling losses conditioned on different levels of ground motion intensity is figured out.
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AUTHORS Zhaoping Jia (1989-), male, the Han
Ziyan Wu (1962-), female, the Han nationality, Ph.D., Professor,
nationality, Bachelor, Master candidate,
Doctoral supervisor, mainly engaged in testing, teaching and
performance-based design and evaluation,
assessment, in 2006, graduated from Northwestern Polytechnical
in 2012, graduated from Northwestern
University in Management Science and Engineering, acquired
PhD. Email: email@example.com
engineering, acquired Bachelor Degree. Email: firstname.lastname@example.org
Qi’ang Wang (1986-), male, the Han nationality, Master, Ph.D. candidate, mainly engaged in structural reliability analysis, in 2013, graduated from Northwestern Polytechnical University in civil engineering, acquired Master Degree. Email: email@example.com
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