pbee literature review

8/15/2019 PBEE Literature Review

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PBEE methodology was primarily introduced to improve the decision making procedures with

respect to seismic performances of buildings. It incorporates the various uncertainties in

seismic analysis procedure. The framework equation is developed on the concepts of

conditional and total probability theorem.

Cornell and Krawinkler (2000) of the Pacific Earthquake Engineering Research (PEER) group

at University of Berkeley, had proposed a unified framework for performance based

engineering in relevance to earthquake engineering hence called PBEE (performance based

earthquake engineering), in which all the four steps discussed below are logically combined

making it suitable to address problems of structural design, assessment of alternative designs,

evaluation of existing structures and retrofit strategies. Mathematically, the framework

translates to (Deierlein et al., 2003)

() = ∭ (|)(|)(|)dλ (2.1) Where λ(x) is the mean annual frequency of exceedance of x and G( | ) is the conditional

complementary distribution function.

Gunay and Mosalam(2013) summarize the four stages of PBEE i.e hazard analysis, structural

analysis, damage analysis and loss analysis. In hazard analysis frequency of exceedance of

intensity of ground motion is estimated. The intensity of ground motion is given the term

intensity measure(IM). Commonly used IMs are the spectral acceleration(SA), spectral

displacement(SD), spectral velocity(SV), peak ground acceleration(PGA), peak ground

displacement(PGD), peak ground velocity(PGV) etc. Probabilistic seismic hazard analysis

(PSHA) is conducted to estimate the frequency of exceedance of an IM. At the structural

analysis stage, thee system response is measured in terms of engineering demand parameter

(EDP). Commonly used EDPs are inter storey drift ratio(IDR), peak displacement, peak shear

force, peak moment etc. The response is found from a series of nonlinear time history analyses

and a plot of EDP vs IM is obtained. Fragility analyses are done to estimate the damage in thestructure. Damage is quantified in terms of damage measure(DM). Both experimental as well

as computational techniques are available to find the fragility curves, which give the

conditional probability of exceedance of a damage level at a given EDP. Finally based on expert

opinions or from previously available data, probability of exceeding a certain loss is found.

Loss is measured in terms of decision variable (DV), which can be either economical i.e the

monetary losses required to restore the structure, downtime, the amount of time after an

earthquake required to bring the structure to fully operational level or in terms of loss in human

lives.


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A closed form expression of the PBEE exists. The closed form solution of exceeding an EDP

has been derived by Jalayer. Jalayer assumes that the hazard curve can be expressed in the

power law form. Jalayer also assumes that the median of EDP at a given IM is log normally

distributed. Based on these two assumptions he develops a closed form expression for

exceeding demand. Based on similar assumptions Mackie and Stojadinovic(2006) continue on

the closed form solution of Jalayer and have found the closed form expression of PBEE, which

is the frequency of exceeding losses in the structure. The closed form expression has been

derived considering both aleatory and epistemic uncertainties in the data.

Mackie and Stojadinovic(2007) also introduced a graphical technique for the estimation of

performance of structures. The graphical technique is also developed based on the assumptions

made in the derivation of closed form expression. The graph is divided into four quarters and

each quarter represents values of different parameters of PBEE i.e the intensity measure,

engineering demand parameter, damage measure and decision variable.

As mentioned by Gunay and Mosalam(2013),the first stage of analysis in PBEE is the hazard

analysis, where the frequency of exceedance of an IM is found. Cornell et al(1979) introduced

one of the earliest ground motion prediction equation(GMPE) to find the hazard curve. Cornell

considered the peak ground acceleration(PGA) to be the IM. He assumes that the PGA follows

log normal distribution whose mean and standard deviation are obtained from the GMPE.

According to him the mean of the hazard curve is dependent on the magnitude and radius of

rupture of the site. He also considers the standard deviation to be independent of the site and

assumes a constant value of 0.57.

Over the years several GMPEs have been proposed and these have been documented in a report

by John Duglas (2011).

A new method called the incremental dynamic analysis (IDA) was introduced by Vamvatsikos

and Cornell (2002) to identify the system response. It involves subjecting the structure to a

series of earthquake records at gradually increasing intensities, thus producing one or more

curves of response vs intensity measure. A unique feature of the IDA curves is its weaving

nature. The weaving nature of IDA curve is due to local softening and hardening.

Patra.P, Bhattacharya.B, (2010) perform single curve IDA for different steel moment resisting

frames considering two different earthquake records, Uttarkashi and Bhuj. They perform IDA

for a simple portal frame, a two storey three bay frame and a three storey two bay frame. Based

on this single IDA drift capacity of the structure is found.

Owing to the computationally intensive nature of IDA, several other methods have been

developed to calculate the system response. One such alternative is the SPO2IDA method


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developed by Vamvatsikos and Cornell (2006). Here the static pushover curve is used to

determine the mean IDA curve. The multi degree of freedom structure is converted to a single

degree of freedom structure and static pushover analysis is done for this structure. The static

pushover curve is idealized as trilinear elastic hardening curve. The drawback of this method

is that it provides only the mean IDA curve and the standard deviation at a given IM is also

unknown.

Kayhani, Azarbakht,Ghafory-Ashtiany (2012) introduced the improved progressive IDA

method to find the building response through the idealization of a multi degree of freedom

structure to a single degree of freedom structure and the utilisation of the genetic algorithm

optimisation technique. They also show that their method significantly reduces the total number

of records required for performing incremental dynamic analysis. The probability of failure

was obtained with this method with ± 15 % error. The major advantage of this method is thatit speeds up the decision making process as the time for analysis is significantly reduced.

Porter K (2006) discusses the general ways of derivation of damage fragility functions using

the damage data. He discusses both experimental as well as analytical methods of obtaining

fragility curves. The most important methods are the actual EDP method, where every

specimen is tested until it fails, the bounding EDP method where only certain specimens are

tested until failure, capable EDP method, where no specimen has failed but their EDPs are

known. Analytically fragility functions are developed using the monte carlo simulation

techniques. In very rare cases fragility functions are developed based on just expert opinion.

Aslani (2005) in his report performs PBEE for buildings and finds the loss curve. He derives

fragility functions from experimental methods. He considers crack width of slab column

connections as damage measure. He considered 4 different damage states based on the width

of the crack. He experimented with 82 specimens and recorded data for each specimen where

the structure exceeded damage state. Based on this exceedance data log normal fragility curves

were fit for individual damage states.

Naess, Gaidai and Karpa(2013) developed the average conditional exceedance rate

method(ACER) to find the probabilities of asymptotes. The approach is based on two separate

components which are designed to improve on two important aspects of extreme value

prediction based on observed data. The first component has the capability to accurately capture

and display the effect of statistical dependence in the data, which opens for the opportunity of

using all the available data in the analysis. A term called the average conditional exceedance

rate is introduced, which is used to find the exceedance probabilities of extreme values.

pbee literature review

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