The paper presents the development of a nonlinear static displacement-based methodology for seismic risk assessment and loss estimation of stone masonry building stock of Pakistan. Experimental investigation of one-third scaled model, tested on shake table, is performed in order to obtain lateral strength and drift limits for stone masonry and develop damage scale for performance-based assessment. Prototype buildings are designed respecting the existing building stock and investigated through nonlinear static and dynamic time history analyses. Nonlinear static mechanical models, for both global and local vulnerabilities, are developed for the considered typology which are used to derive analytical structure-dependent fragility functions considering expected sources of uncertainties explicitly in contrary to the conventional procedures. Furthermore, seismic risk assessment is performed for different scenario earthquakes and presented in terms of structure-independent fragility functions to estimate the mean damage ratio, the repair cost as a fraction of replacement cost, and casualties, with the dispersion being quantified, given source-to-site distance and magnitude for an earthquake event. The methodology is tested for seismic risk assessment of the considered typology in recent 2005 Kashmir earthquake, which is reasonably predicted. Future development of the methodology is required with additional experimental tests on rubble stone masonry material in order to increase confidence in future applications.
Stone masonry buildings constitute a substantial portion of urban and rural building stock of northern areas of Pakistan. Two wythes random rubble stone masonry walls (Figure
Typical rubble stone masonry building walls, existing buildings, and new structural schemes (a) and spatial distribution of stone masonry buildings in Pakistan [
Such building systems have shown very poor performance in past earthquakes and lead to huge losses of life and economy. Collapse of such structures featured prominently in the 2005 Kashmir earthquake [
Seismic risk assessment and loss estimation of structures have developed from initial studies, in the past 30 years, which used empirical approaches to predict the socioeconomic impacts of earthquakes, to more recent methodologies based on numerical predictions of the capacity of buildings to resist strong ground shaking [
The methodology makes use of analytical fragility functions and a mathematical model for building stock, which is represented as a nonlinear static single degree of freedom (SDOF) system defined completely by secant vibration period, displacement capacity, and viscous damping, to assess the seismic risk and socioeconomic impacts of a given earthquake event. The seismic demand on building stock is defined by 5% damped displacement response spectrum using code-based (for intensity-based), GMPE-based (for scenario earthquakes), and/or UHS-based (for estimation of loss exceedance curve and annualized losses) spectrum representation depending on the scope of the study. GMPE stands for ground motion prediction equation and UHS stands for uniform hazard spectra. To better understand the methodology, it is depicted graphically (Figure
Graphical representation of DBELA: idealized displacement response spectrum (a), analytical fragility functions, global mechanism (b), and damage scenario (c).
In this methodology, the first step is to generate elastic displacement response spectrum for a given earthquake, which is overdamped in an iterative fashion to compute the structure-specific seismic demand (seismic intensity). This seismic intensity is used to compute the number of buildings in different damage states from the analytical fragility functions in order to develop damage scenarios. The number of buildings in different damage states is transformed to mean damage ratio (MDR), which quantifies the regional seismic risk and gives an estimate of direct economic losses for the considered earthquake. The conversion of building damage states to MDR is performed using earthquake loss model [
For seismic assessment, the stone masonry building stock of Pakistan is broadly classified in two major classes based on the predominant seismic response mechanism during an earthquake. Masonry buildings provide seismic resistance to ground shaking through the development of in-plane/out-of-plane forces/deformations in structural walls. The buildings that have enough structural integrity provide seismic resistance predominantly through in-plane shear/flexure response of structural walls (global mechanism) while buildings with less structural integrity respond primarily in out-of-plane response of walls (local mechanism). The methodology uses SDOF systems to assess the seismic performance of each class of building stock. The SDOF system, called a mechanical model, has nonlinear lateral force-displacement response to assess the seismic performance of structures. The mechanical model simulates the response of the structural system in terms of its displacement capacity, energy dissipation, and secant vibration period for seismic assessment. The SDOF system derivation for each class of buildings are discussed as follows.
The seismic response of stone masonry buildings having rc slab, well-connected orthogonal walls, and ring beams is mainly governed by the global response of buildings, because of the in-plane integrity of walls provided by the rc slab, with shear failure dominated mechanism of in-plane walls [
Nonlinear static SDOF idealization, mechanical model, of stone masonry buildings for global mechanism.
In this figure,
For seismic assessment, the mechanical model is completely defined by secant vibration period, limit state displacement capacity, and energy dissipation characteristics of building represented as viscous damping:
Stone masonry buildings without rc slabs and ring beams having orthogonal walls not properly connected, or due to loss of in-plane walls integrity during seismic excitations, respond in local out-of-plane collapse of portion or complete walls [
Nonlinear static SDOF idealization, mechanical model, of masonry wall for out-of-plane mechanism, modified.
In Figure
The resistance of local out-of-plane mechanism to earthquake excitation is governed by the wall geometry, boundary condition, self-weight, and precompression level of rocking portion of wall while being less affected by the masonry material properties [
Observed out-of-plane mechanism of masonry buildings [
Fragility functions, also called fragility curves, vulnerability curves, and/or building damage functions, describe the number of buildings reaching or exceeding a given damage state given the seismic intensity, represented as peak ground shaking parameters or spectral quantities [
For loss estimation on regional scale, the uncertainties and variabilities in structural characteristics, geometrical and material uncertainties, can be obtained through on site survey of building stock and laboratory investigation of structural materials. The survey can better provide information on the likelihood of different geometrical features of regional building stock, for example, beam/column depth, width, length, reinforcement details, and number of structure’s storeys [
The DBELA methodology makes use of displacement-based analytical fragility functions where the seismic intensity is defined as a vector-based inelastic displacement demand on the structural system due to its direct correlation with the structural expected performance level for a given seismic demand [
Flow chart for the derivation of displacement-based analytical fragility functions, global mechanism.
Flow chart for the derivation of displacement-based fragility functions, local mechanism.
The first step of the method is the generation of random population of buildings which represent a given class of building typologies within a given urban/rural exposure. Controlled Monte Carlo simulation is used to generate thousands of buildings, each with different geometrical and mechanical properties being defined using a complete probabilistic distribution with prescribed mean and coefficient of variation.
The second step of the method is to define random seismic demand on the generated buildings which is performed through the use of random linear 5% damped displacement response spectrum. Special variability of ground motions is not considered in the fragility derivation which can be considered later in the application of fragility functions for developing damage scenarios for regional risk assessment and loss estimation.
For each of the spectra, (a) for each limit state the secant vibration period, displacement capacity, and viscous damping of the buildings from the random populations are computed using calibrated structure-specific mathematical models, (b) the displacement demand on each of the buildings is obtained from the overdamped displacement response spectrum at the limit state vibration period of that building which is then compared with the displacement capacity of the building to predict its performance, and (c) the number of buildings having capacity less than the demand is summed and divided by the total number of the generated buildings to obtain the probability of exceedance of a given limit state. Similar hypotheses are used by other support methodologies [
For each of the spectra, (a) the median yield vibration period and median yield displacement capacity are obtained from the generated building stock, (b) the spectral displacement demand at the median yield vibration period is obtained from the 5% damped elastic displacement spectrum, (c) this is compared with the median yield displacement capacity, if in case the demand is less than the yield displacement (i.e., this defines the median spectral displacement demand on the structures), and (d) for spectral displacement demand greater than the median yield displacement, the performance point is obtained in an iterative fashion which defines the median spectral displacement demand (seismic intensity).
The probability of exceedance for each limit state is plotted versus the median displacement demand (seismic intensity) for each of the random spectra. Available cumulative distribution functions are fit to the data and the unknowns of the functions are obtained to completely describe the fragility functions for future applications.
The first step of the method is the generation of random population of buildings which represent a given class of building typologies within a given urban/rural exposure. Controlled Monte Carlo simulation is used to generate thousands of buildings, each with different geometrical and mechanical properties being defined using a complete probabilistic distribution with prescribed mean and coefficient of variation.
The second step of the method is to define random seismic demand on the generated buildings which is performed through the use of random linear 5% damped absolute displacement response spectra [
For each of the absolute spectra, (a) vibration period, displacement capacity, and viscous damping of the buildings from the random populations can be computed for a given local mechanism at the collapse limit state using simplified empirical models, (b) the displacement demand on each of the buildings for local mechanism can be obtained from the 5% damped absolute displacement response spectrum at the global secant vibration period of building which is then amplified with the out-of-plane amplification spectrum/function at the vibration period of the considered local mechanism in order to compare the demand with the displacement capacity to predict its performance, and (c) the number of buildings having capacity less than the demand is summed and divided by the total number of generated buildings in order to obtain the probability of exceedance of local collapse mechanism.
For each of the absolute spectra, (a) the yield vibration period for global response and median secant vibration period for local response are computed and (b) the spectral displacement demand at the median yield vibration period of the given class is obtained from the 5% damped absolute displacement spectrum which is amplified with the floor amplification function at the local mechanism’s secant period in order to obtain the median spectral displacement demand (seismic intensity).
The probability of exceedance for each limit state is plotted versus the median displacement demand (seismic intensity) for each of the random spectra. Available cumulative distribution functions are fit to the data and the unknowns of the functions are obtained to completely describe the fragility functions for future applications.
The DBELA assessment approach needs the basic material properties of masonry and geometric detailing to develop structure-specific mechanical models (in-plane/out-of-plane) for the assessment of global and local capacity of masonry structural systems. The essential material properties, compressive strength, elastic moduli, tensile strength, and so forth, of considered rubble stone masonry cannot be obtained reliably at the section level due to the difficulties in performing tests on masonry prism, reproducing the true replica of the filed condition and huge scatter in the observed behavior, which is due to the fact that when stone-to-stone contact is found during the compression tests, very huge value of compressive strength is achieved which gets minimal value when stone-to-void possibility is found. Huge uncertainties in the material properties at the section level make the global response less reliable to be achieved. Also, the material properties at section level cannot directly provide guidance on the performance levels, damage scale, and predominant seismic response mechanism of a building. Thus, a full reduced scaled structural model is tested to obtain the global capacity parameters and develop damage scale in order to develop the DBELA tool for global seismic risk assessment of stone masonry buildings with rc slab in Pakistan.
One-third scaled structural model of single storey and single room is designed for corresponding prototype structure using the similitude principles (Figure
Details of the stone masonry structural model tested on shake table, plan view, elevation and actual model.
The structural model is subjected to 5% scaled Kobe record with linear amplitude scaling until the total collapse of structural model. The model is inspected for the observed damage after every run. The floor acceleration and displacements, on each in-plane wall and at the base of the model, are recorded and processed. The floor acceleration is normalized by the seismic mass of the structural model in order to obtain the base shear for the corresponding equivalent SDOF system. Also, the average floor displacement demand is obtained which is corrected with the base displacement demand in order to obtain the relative displacement demand on the system, which is normalized by the model height to compute the corresponding drift demand on the system. The observed physical damage states of tested structural model are shown (Figure
Capacity parameters and damage states of the corresponding prototype system.
Damage level | Equivalent base shear | Drift demand | Damage description |
---|---|---|---|
Minor (DS1) | 1.67 | 0.05 | Separation of reinforced concrete slab and ring beams from walls |
Moderate (DS2) | 3.92 | 0.48 | Crack initiation in the masonry in-plane walls and around the openings |
Major (DS3) | 5.00 | 1.49 | Widening of cracks and falling of stones from the out-of-plane walls |
Collapse (DS4) | 1.47 | 2.65 | Complete collapse of the structural model |
Observed damage states of tested structural model, from (a) to (d): DS1, DS2, DS3, and DS4.
Stone masonry buildings are practiced abundantly in the northern areas, urban and rural, of Pakistan due to large local availability of stone material and low cost of labor in construction. Stone masonry with earthen floor, wooden/steel truss with GI sheet, and/or rc floor are the common residential building construction practice for single storey in rural areas and up to three storeys with rc floors in urban areas. Stone masonry in cement mortar contributes 50% in overall to the total building stock in the recent earthquake affected areas (http://www.erra.pk/). The most prevailing building dimensions range from 8 m × 5 m to 15 m × 5 m with typical wall density, with the ratio of the cross-sectional area of the in-plane walls to the total floor area, ranging from 10% to 15% [
Due to unavailability of basic material properties for stone masonry, and the reasons mentioned earlier, it is not straightforward to develop numerical tools for future applications for global analysis. Thus, the present study considered the simplified equivalent frame approach, SD-SAM [
Equivalent frame idealization of masonry wall (a) and nonlinear force-displacement response of frame elements (b) [
In the first step, a generic prototype of equivalent frame is generated for the tested structural model and designed, respecting the geometric detailing and loading conditions, with the available empirical shear strength models for masonry walls, previously developed [
In the second step, the tensile strength,
Comparison of the observed and predicted lateral capacity of tested stone masonry building (a) and phenomenological nonlinear hysteretic response of frame element (b).
The present study considered 125-case study 2D two-storey structural models, selected with different floor area, wall density, and material properties, in order to take into account the regional geometric and material uncertainties explicitly in the capacity evaluation, designed with the material sectional properties obtained in the previous section. The frame elements are assigned with bilinear Takeda type rule [
Mean of the acceleration spectra considered for NLTHA and details of the accelerograms used in the study.
The accelerograms are linearly scaled in order to observe the postyield response of the models which is used then to derive static SDOF system, mechanical models, for the considered building typology in order to retrieve the dynamic characteristics of buildings in preyield and postyield limit states and develop empirical models for limit state’s vibration period and displacement capacity. It is worth mentioning that the numerical tool used herein is simplified but respecting the fundamental structure’s parameters, that is, stiffness, strength, ductility, and energy dissipation, sufficient to demonstrate the global structural performance for seismic loading. Also, the findings of numerical tool are used for the risk assessment of buildings on regional scale where, apart from large case study structural models, the regional structural geometric and mechanical properties are approximately defined which makes the present tool as a reasonable choice for structural analysis.
The scope of the dynamic analysis is to compute the equivalent base shear and equivalent displacement demand at different limit states of masonry walls on the ground floor, since ground storey mechanism is governed for the considered typology, using the proposed SDOF derivation [
Period coefficient for different wall density and floor areas (a) and for all cases study structural models (b).
It is observed that for a given structural system the period reduces with increasing wall density. Similar trend of vibration period is also observed for increasing floor area. The period coefficient obtained for all the cases study structural models, 1250 cases, takes into account the geometric (floor area and wall density) and material uncertainties besides the variability introduced by earthquake excitations. Existing and conventional code-based formulae considered
Additionally, the structural models are analyzed to develop the limit states displacement capacity model (
Limit states displacement profiles of a two-storey structural model (a); displacement coefficients at yielding (b) and near collapse limit state (c).
The proposed methodology is used herein to derive analytical fragility functions for urban rubble stone masonry buildings of Pakistan. Application of the methodology is also carried out for brick masonry urban building stock of NE Pakistan [
Controlled Monte Carlo simulation is used to generate random building stock with different geometric and mechanical properties considering lognormal probability density function (pdf) for all the parameters involved in the capacity evaluation (Table
Parameters used in the random generation of stone masonry buildings of Pakistan.
Parameter | Mean value | C.O.V. | Lower bound | Upper bound | Probability distribution type |
---|---|---|---|---|---|
| 3.00 | 30.00 | 2.50 | 3.50 | Truncated lognormal |
| 2.50 | 30.00 | 2.00 | 3.00 | Truncated lognormal |
Drift (%) | |||||
LS1 | 0.240 | 16.10 | 0.16 | 0.33 | Truncated lognormal |
LS2 | 0.350 | 15.08 | 0.24 | 0.50 | Truncated lognormal |
LS3 | 0.768 | 30.00 | 0.53 | 1.10 | Truncated lognormal |
LS4 | 1.016 | 30.00 | 0.70 | 1.45 | Truncated lognormal |
| 0.727 | 01.15 | 0.69 | 0.75 | Truncated lognormal |
| 0.807 | 14.22 | 0.44 | 1.23 | Truncated lognormal |
| 0.400 | 20.00 | 0.28 | 0.50 | Truncated lognormal |
| 400 | 20.00 | 300 | 500 | Truncated lognormal |
| 0.8 | 10.00 | 0.50 | 1.00 | Truncated lognormal |
| 0.75 | 16.00 | 0.60 | 0.90 | Truncated lognormal |
Stone masonry buildings with rc slab did not show prominent out-of-plane failure in experimental test and during 2005 Kashmir earthquake but were damaged mainly due to diagonal cracks in wall, cracking around the corners of opening, horizontal cracks at the roof level, and complete collapse of buildings [
Once the regional building stock is generated and the limit state capacities are evaluated in a probabilistic fashion, random linear 5% damped displacement response spectra are generated. The global in-plane assessment is performed through overdamping the linear spectrum, using overdamping factor [
Definition of seismic demand on the out-of-plane walls: absolute spectrum (a) and out-of-plane amplification spectrum (b).
The absolute spectrum is anchored at the peak ground displacement (pgd) and linearly increased to maximum displacement demand at corner period following the recommendation of Priestley et al. [
Analytical fragility functions for two-storey stone masonry building stock of Pakistan: in-plane global mechanism (a) and out-of-plane local mechanism (b).
It is mentioned that fragility functions are needed to be developed in terms of structure-dependent vector-based intensity measures (inelastic displacement demand in the present case which is well correlated with the structural expected performance) and in order to avoid the problem of being specific to given assumed spectral shape and region, for example, code spectra and UHS for a given site which are not conceptual [
Thus fragility functions are derived for the MDR of stone masonry buildings of Pakistan expected at a site for a given scenario earthquake. Different possibilities of source-to-site distance and magnitude are selected, considering soft soil condition (type D of NEHRP soil classification as recommended for Pakistan [
Charts for MDR of stone masonry buildings for scenario earthquakes.
As expected from the structural mechanics standpoint, the local out-of-plane mechanism results in relatively higher seismic risk and losses as compared to global in-plane mechanism of structural systems. Depending on the site building characteristics, if buildings are provided with rc slab and ring beams (like the present situation in the earthquake affected areas, http://www.erra.pk/, and some urban exposure) only the global mechanism’s charts have to be used to compute the regional seismic risk and losses for a given scenario earthquake, described only by the source-to-site distance and moment magnitude. If the regional buildings do not meet the minimum criterion to ensure the in-plane integrity of structural system and global seismic response mechanism [
The expected causality rate is presented in terms of the occupancy in a housing unit and the total number of building stocks in the region. Thus the number of housing units in a region and the likelihood function of occupancy per housing unit, 7 to 12 for the considered typology [
Charts for expected casualty rate in collapsed stone masonry buildings for scenario earthquakes, in-plane mechanism.
Charts for expected casualty rate in collapsed stone masonry buildings for scenario earthquakes, out-of-plane mechanism.
The structure-independent fragility functions are used to assess the seismic risk of stone masonry buildings for a scenario earthquake that recently occurred in the country, that is, 2005 Kashmir Earthquake. The Kashmir earthquake has a moment magnitude of 7.6 that struck most of the northern areas of Pakistan on 8 October 2005 at 08:50 AM local time (
Almost all the buildings, mainly stone and block masonry laid in cement sand mortar with rc slabs or GI sheet roofing, collapsed in the areas close to the epicenter [
Chart for MDR of stone masonry buildings of Pakistan for scenario 2005 Kashmir earthquake.
Using only the in-plane fragility function gives an estimate of the MDR equal to 0.26 which gives an error of about +67.09 percent thus underestimating the risk, and the out-of-plane fragility function gives estimate of MDR equal to 1.00 which gives an error of about −26.58 percent thus overestimating the seismic risk, and the combined fragility function gives an estimate of MDR equal to 0.85 which gives an error of about −7.60 thus overestimating the site risk. Nevertheless, the observed MDR is reasonably predicted in case of combined fragility function. Also, the proposed fragility functions are conservative, that is, giving slightly overestimated seismic risk and losses.
The mean prediction for seismic risk can reasonably predict the observed response during earthquake, as observed in the previous section. Nevertheless, considerable uncertainties can result in loss estimation for a given region considering a single scenario earthquake [
Trends in COV and standard deviation versus MDR for stone masonry building (global mechanism) of Pakistan.
It is observed that COV tends to decrease with increase in MDR, a similar trend also observed elsewhere [
Seismic risk assessment of stone masonry buildings of Pakistan is performed using an innovative and state-of-the-art nonlinear static methodology for seismic risk assessment and loss estimation of buildings on regional scale. The methodology takes different possible sources of uncertainties in an explicit and transparent manner to compute the structural capacity and assess the seismic performance of structural system. Analytical displacement-based fragility functions are derived for the considered buildings, for both local and global vulnerabilities, which are used to estimate the seismic risk and losses for scenario earthquakes and develop structure-independent fragility functions. Simplified charts are provided to compute the socioeconomic impacts of earthquakes given the magnitude and source-to-site distance of the event. Comparison of the methodology and derived fragility functions is performed with the recent earthquake observation, in predicting seismic risk of the considered typology, which is reasonably predicted. The methodology can provide also estimate of uncertainties in regional seismic risk to provide help in decision-making. However, from the derived structure-independent fragility functions it is possible to develop seismic risk prediction equations for regional loss estimation. Comparison with large earthquake observation databases can provide estimate of inter- and intraevent uncertainties in seismic risk prediction in similar fashion as ground motion prediction equations do. It will make the regional loss estimation studies to assess the regional risk directly using the derived equations, once developed for a region. Similar studies will be performed on other regional building stocks of Pakistan once more experimental and reliable data are made available for the region. Additional data on stone masonry material and buildings can further improve the findings herein in order to increase confidence in future applications. The findings herein are the first of its kind for the considered region.
The research work presented herein is the further extension of the methodology, adopted for the fragility functions derivation under the EU funded project SYNER-G (Systematic Seismic Vulnerability and Risk Analysis for Buildings, Lifelines Networks and Infrastructures Safety Gain), to derive functions for the direct seismic risk assessment of buildings given the scenario earthquake.
The authors declare that there are no conflicts of interest regarding the publication of this paper.