Seismic risk evaluation of historic masonry structures in an Algiers district, Algeria, using fragility and vulnerability functions: A case study of Belouizdad district

Vulnerability index (VI) method

2.1

The VI method assigns a numerical value to each surveyed building, reflecting its expected seismic performance. The VI is calculated as the sum of coefficients k_i that represent the level of vulnerability associated with 12 structural and non-structural parameters known to influence seismic behavior. Similar studies have been carried out for reinforced concrete [40], reinforced masonry [41], and steel buildings [42].

In this study, 12 parameters are considered (Table 1). Each parameter coefficient k _i is classified into four categories:

• C1: compliant with current Algerian seismic code,

• C2 and C3: compliant with earlier Algerian codes,

• C4: unsafe configuration.

The VI is given by

Table 1 presents the weighting parameters, which were derived from statistical surveys and empirical data from past Algerian earthquakes [36,37,38,39].

To validate the parameter selection, correlation coefficients were computed between the total VI and the partial indices of each parameter, calibrated using observed damage from the Ain Temouchent (1999) and Boumerdes (2003) earthquakes [39]. Table 2 shows that wall resistance, connections, soil conditions, and general maintenance are the most influential parameters. Less influential factors (e.g., pounding, architectural details) were retained for methodological consistency with international frameworks such as Benedetti & Petrini (1984) and Risk-UE.

This approach relies primarily on visual survey parameters for two reasons:

1. The large building stock (643 structures) makes in situ mechanical testing infeasible on the urban scale.

2. The VI methodology has been widely applied in Europe and the Mediterranean under similar constraints, with robust results.

Calibration of the weighting coefficients against post-earthquake damage observations (Ain Temouchent 1999, Boumerdes 2003) strengthens the validity of the approach. While no new in situ mechanical testing was performed for Algiers, the integration of empirical calibration data provides indirect validation. Future work will combine survey-based assessments with targeted in situ testing to refine parameter weights for the Algerian context.

Finally, three vulnerability classes are defined (Table 3): Green, Orange, and Red, corresponding to low, medium, and high vulnerability.

Study area

2.2

The investigation was conducted in the Belouizdad district of Algiers, Algeria (Figures 1 and 2). This coastal commune covers an area of 2.16 km² with a population density of 20,394 inhabitants/km². The building stock includes 643 masonry structures, primarily stone or brick walls (average thickness ≈60 cm) with vaulted floors. Most of these buildings date from the colonial period (1830–1962), during which urban and architectural practices evolved through four phases (1830–1854; 1854–1881; 1881–1915; and 1915–1962) [36].

Figure 1

Source: Author’s contribution.

Figure 2

Source: Author’s contribution.

All buildings in the municipality were surveyed using the data sheet developed by Djaalali [36]. The collected data were digitized and managed in a Geographic Information System (GIS) to facilitate organization, visualization, and interpretation of the results (Figure 3).

Figure 3

Source: Author’s contribution.

The building heights range from one to seven stories. Their distribution is given in Table 4 and illustrated in Figure 4. Two-story buildings are predominant, representing 34.2% of the total stock. One- and three-story buildings account for about 20% each. Together, these categories comprise more than 70% of the masonry buildings, while the buildings with other stories are much less common.

Figure 4

Source: Author’s contribution.

Data regarding construction date were available for only 138 buildings (≈21% of the stock). These were classified into four periods (Table 5, Figure 5). Although partial, tests confirmed that this sample is representative: its distribution by number of stories and maintenance condition matches that of the full dataset. Moreover, since construction date is only 1 of the 12 VI parameters, sensitivity analysis showed that excluding it alters the mean VI by less than 0.03. Thus, partial coverage has negligible influence on the results.

Figure 5

Source: Author’s contribution.

Figure 6 shows that fewer than 2% of buildings remain in their original state. More than 50% have undergone moderate modifications, and about 20% major alterations, factors that significantly increase vulnerability.

Figure 6

Source: Author’s contribution.

The VIs of all 643 buildings were calculated and mapped in GIS (Figure 7). Results (Figure 8) show that about 80% of the buildings fall into the medium vulnerability class (VI = 0.20–0.60), while approximately 9% are highly vulnerable (VI > 0.60). In total, around 89% of Belouizdad’s masonry stock is classified as seismically vulnerable, primarily due to age, poor maintenance, and structural alterations.

Figure 7

Source: Author’s contribution.

Figure 8

Source: Author’s contribution.

Vulnerability function

2.3

Vulnerability functions describe the expected mean damage ratio (MDR) as a function of seismic intensity. In this study, Algerian vulnerability functions were derived from those developed for Friuli (Italy) using the translation methodology of Huo et al. [14]. Both regions share similar URM typologies – low- to mid-rise masonry with weak mortar, irregular layouts, and poor wall connections – as well as comparable geological conditions (sedimentary soils with amplification effects).

The translation applies a horizontal shift of ΔMMI = 0.12 along the intensity axis to account for differences in seismic provisions and construction practices between Friuli and Algiers. This adjustment was calibrated using empirical data from the 2003 Boumerdes earthquake (Mw 6.8, PGA ≈ 0.27 g). Observed MDR values for URM buildings in Algiers during this event (0.20–0.23) closely matched the translated predictions (MDR ≈ 0.21), supporting the validity of the shift. Nevertheless, Algerian URM differs from its European counterpart in workmanship, maintenance, and occupancy, which may shift MDR by ±10–15%. This justifies the more conservative character of Algerian functions.

Vulnerability functions are expressed as relationships between Modified Mercalli intensity (MMI) and MDR (%). They depend on the VI, which ranges from 0 (no damage) to 1 (collapse). For this study, the VI was divided into ten classes (CL1: 0.0–0.1, CL2: 0.1–0.2, …, CL10: 0.9–1.0) to capture the variability of masonry performance (Figure 9). CL1–CL2 correspond to low vulnerability, CL3–CL6 to medium vulnerability, and CL9–CL10 to high vulnerability. The resulting Algerian vulnerability curves are shown in Figure 10.

Figure 9

Source: Author’s contribution.

Figure 10

Source: data from Djaalali et al.

An analytical representation of these curves was derived by regression, establishing a functional relationship between MDR, VI, and intensity (I):

The analytical expression of the MDR as a function of intensity and VI (equations (2) and (3)) was derived by regression fitting of the translated vulnerability curves, following the methodology established by Djaalali et al. [39]. The functional form was selected because it provides a good fit to the empirical vulnerability functions obtained from the Friuli–Algiers translation, while ensuring monotonicity and stability across the full VI range. Although the equation appears complex, it is simply a regression-based interpolation formula that captures the nonlinear dependence of MDR on both intensity and VI. Its coefficients were calibrated to minimize residuals between the analytical curve and the empirical points (R ² > 0.95). In this study, the same functional form was adopted to ensure consistency with previous European applications. The coefficients were recalibrated specifically for Algerian URM stock using the translation procedure described above, ensuring that the analytical representation is not arbitrarily chosen but statistically validated.

Finally, it is important to distinguish between vulnerability and fragility curves:

• Vulnerability curves describe the expected MDR as a function of seismic intensity.

• Fragility curves express the probability of exceeding specific damage states at given intensities.

Both are developed in this study: vulnerability curves (Section 2.3) quantify mean MDR values, while fragility curves (Section 2.4) provide probabilistic damage-state exceedances.

Fragility curves assessment

2.4

Fragility curves estimate the probability of reaching or exceeding a specific damage state as a function of seismic intensity. They are typically modeled by log-normal functions, which capture variability in damage among buildings of the same type [43].

Damage measure and performance level

2.4.3

Defining a consistent measure of seismic damage is a key step in fragility analysis. For URM buildings, FEMA 356 defines three performance levels, while HAZUS [29] uses four limit states (slight, moderate, extensive, collapse). In this study, the Park & Ang thresholds were adopted to ensure comparability with existing European studies, and fragility curves were developed accordingly.

Figure 11 presents the empirical MDR data points for different seismic intensities together with their fitted log-normal fragility curves. Following Park et al. [28], the log-normal distribution provides an appropriate statistical description of building response. The distribution parameters – the log-normal mean (µ _ln D) and the standard deviation (σ _ln D) – were estimated from probability plots of the empirical data.

Figure 11

Source: Author’s contribution.

The log-normal mean value and the standard deviation are estimated from the y-intercept and the slope of the fitted line, respectively. The log-normal parameters for description of fragility curves are given in Table 7.

Table 7

Log-normal parameters.

MMI	μ ln D	σ ln D
VII	2.444	1.886
VIII	−1.537	1.335
IX	−1.062	1.182
X	−0.764	0.881

Source: Author’s contribution.

The larger dispersion at lower intensities (e.g., σ _ln D = 1.886 at MMI VII) reflects greater variability in damage onset, while dispersion decreases at higher intensities (σ _ln D = 0.881 at MMI X), where severe damage becomes more consistent across the building stock.

Construction of fragility curves

3.1

The probability of exceeding a given damage state was calculated using the log-normal cumulative distribution function (CDF) (equation (5)), where Ф(.) denotes the standard normal CDF The adopted damage thresholds correspond to those of Park & Ang (1985): None = 0.01, Light = 0.10, Moderate = 0.20, Severe = 0.50, and Collapse = 0.85. A sensitivity analysis of these thresholds is reported in Section 2.4.2, confirming their robustness.

Fragility curves for Algiers URM buildings are presented in Figure 12. They show the probability of exceeding each limit state as a function of seismic intensity. The log-normal fitting was validated through goodness-of-fit metrics (Section 3.4), confirming its adequacy for Algerian URM stock.

Figure 12

Source: Author’s contribution.

As an illustration, for an earthquake of intensity MMI IX, the fragility curves indicate that the probability of exceeding the moderate damage state (MDR = 0.20) is approximately 68%, while the probability of exceeding the severe damage state (MDR = 0.50) is about 38%. The likelihood of exceeding the collapse state (MDR = 0.85) remains significant at roughly 22%. These results highlight the limited seismic capacity of Algiers’ URM buildings, where even moderate-to-strong ground motions are likely to cause substantial structural damage.

Validation of fragility curves

3.2

Within the framework of the European program RISK-UE, the LM1 method was developed to assess the seismic vulnerability of structures in Europe. This approach correlates macroseismic intensity with observed earthquake damage, expressed through DPM derived from the European Macroseismic Scale (EMS-98). The LM1 method distinguishes between no damage and five levels of damage (slight, moderate, significant, very significant, and destruction). The Building classification matrix defines 23 structural classes grouped by (i) structural system and (ii) building material. The framework introduces the VI as a quantitative measure ranging from 0 (no damage) to 1 (collapse), and provides a semi-empirical function to correlate the average damage level (μ _D) with macroseismic intensity (I) and VI.

In this study, the comparison focuses on URM buildings, which in Algeria are predominantly constructed of stone and/or brick masonry with vaulted floors. These correspond to the M1.2 and M3.3 typologies in the Risk-UE classification, with mean VIs of 0.74 and 0.70, respectively (Table 8).

Figure 13 compares the Algerian fragility curves (log-normal distribution) with the Risk-UE fragility curves (beta distribution). The Algerian curves consistently lie above their European counterparts for moderate, severe, and collapse damage states, indicating higher probabilities of exceedance at the same intensity. Average deviations are summarized in Table 9.

Figure 13

Source: Author’s contribution.

These differences are partly attributable to the use of different probability models (log-normal vs beta), but they also reflect intrinsic characteristics of Algerian URM buildings – namely, poor maintenance, informal modifications, and higher occupancy loads – that reduce their seismic performance.

Crucially, validation was not limited to comparison with Risk-UE functions. Indirect calibration was performed using Algerian earthquake observations. Although detailed building-by-building datasets are unavailable for Algiers, the weighting coefficients of the VI (Table 1) were calibrated against observed damage correlations from the Ain Temouchent (1999) and Boumerdes (2003) earthquakes [39]. This provides empirical support for the parameterization of the VI and, by extension, for the derived fragility curves.

In summary, while direct calibration with detailed local datasets remains an open challenge, the fragility curves developed here are supported by both international benchmarking (Risk-UE) and indirect validation using Algerian seismic experience. Future research should incorporate more comprehensive post-earthquake surveys to further strengthen calibration and reduce epistemic uncertainties.

Seismic scenarios

3.3

Seismic scenarios were simulated for the Belouizdad district under intensities ranging from MMI VII–X, using the fragility curves developed in this study together with the analytical MDR functions (equations (2) and (3)). The resulting damage distributions for the 643 surveyed URM buildings are illustrated in Figures 14–17.

Figure 14

Source: Author’s contribution.

Figure 15

Source: Author’s contribution.

Figure 16

Source: Author’s contribution.

Figure 17

Source: Author’s contribution.

Key outcomes are as follows:

• MMI VII: more than 30% probability of exceeding moderate damage, ≈18% probability of exceeding severe damage, and >10% probability of collapse.

• MMI VIII: ≈75% of buildings are expected to reach moderate damage (probability = 0.52), while ≈2% are predicted to reach severe damage.

• MMI IX: about 90% of buildings are expected to exceed the moderate damage state (probability = 0.68), with ≈10% reaching severe damage.

• MMI X: ≈69% of buildings are expected to experience at least moderate damage (probability = 0.83), and ≈31% are predicted to reach severe damage (probability = 0.47).

These results confirm the low seismic performance of Algiers’ URM stock, which is largely attributable to construction practices predating seismic codes, cumulative deterioration, and widespread informal modifications.

It is important to note that the exceedance probabilities reported here represent median values. Accounting for the dispersion parameters of the fragility curves (σ _ln D, Table 7), uncertainty ranges are on the order of ±10–15%. For example, the estimated 0.68 probability of exceeding moderate damage at MMI IX could realistically vary between 0.58 and 0.77. This probabilistic framing underscores the need to interpret the results as ranges rather than deterministic values.

From a risk management perspective, the implications are considerable. With a population density exceeding 20,000 inhabitants/km², an MMI IX earthquake – where ≈90% of URM buildings are predicted to suffer at least moderate damage – could displace several thousand residents. Using an average replacement cost of 250–300 €/m², direct physical losses for MMI IX–X events could plausibly reach tens of millions of euros. While a full socioeconomic impact model is beyond the scope of this study, these order-of-magnitude estimates demonstrate the practical relevance of the fragility functions for guiding preparedness, prioritizing retrofitting, and informing urban resilience policies in Algiers.

Validation of the log-normal distribution assumption

3.4

Fragility curves are most commonly modeled using log-normal CDFs because of their simplicity, statistical tractability, and widespread adoption in seismic vulnerability studies. To evaluate whether this assumption is appropriate for Algerian URM buildings, alternative probability models (Weibull and Gamma distributions) were also tested.

Goodness-of-fit was assessed using standard statistical criteria, including the Kolmogorov–Smirnov (KS) test and information criteria (Akaike information criterion, AIC; Bayesian information criterion, BIC). Across most damage states, the log-normal distribution consistently produced the lowest AIC and BIC values and passed the KS test with p-values above 0.05, indicating no significant deviation from the empirical fragility data. For the lowest damage state (DS1), the Weibull distribution yielded slightly lower AIC and BIC values, suggesting a marginally better fit. However, this improvement was not statistically significant at the 5% level. For reasons of consistency with the Risk-UE framework and comparability with previous international studies (e.g., Park & Ang, 1985), the log-normal distribution was therefore retained as the primary model.

This statistical validation supports the robustness of the log-normal model for fragility curve construction in the Algerian context, while acknowledging minor limitations at low damage thresholds.

A notable outcome of this study is that the Algerian vulnerability and fragility curves are systematically more conservative than their European counterparts. This conservatism is empirically supported by field observations: more than half of the Belouizdad masonry stock has undergone medium to major structural modifications, and fewer than 2% of buildings remain in their original state (Figure 6). Poor maintenance conditions – identified as one of the strongest correlating parameters with overall vulnerability (r = 0.46–0.53, Table 2) – are also widespread. Together with informal alterations and high occupancy loads, these conditions reduce the seismic performance of Algerian URM relative to European analogues. As a result, the conservative nature of the Algerian fragility functions is not only theoretically justified but also firmly grounded in empirical evidence from the local building stock.

Uncertainty analysis

3.5

Building on the typological limitations outlined in Section 3.4, this section discusses the main sources of uncertainty in the methodology and their potential impact on the derived vulnerability and fragility functions. Although the results presented here are expressed deterministically, both epistemic and aleatory uncertainties must be acknowledged:

• VI method. Parameter scoring is based primarily on visual surveys, which may introduce subjectivity. Calibration against observed damage from the Ain Temouchent (1999) and Boumerdes (2003) earthquakes helps mitigate this effect, but variability among surveyors and incomplete structural documentation remain potential sources of error.

• Translation of vulnerability functions. The Friuli → Algiers intensity shift (ΔMMI = 0.12) was derived empirically to capture differences in workmanship and maintenance. Sensitivity tests suggest that this adjustment introduces an uncertainty of approximately ±10–15% in the MDR.

• Analytical MDR formulation. The regression-based equations (equations (2) and (3)) achieve high coefficients of determination (R ² > 0.95) but still carry residual errors. These residuals may propagate into fragility estimates, particularly at higher intensity levels.

• Fragility curve fitting. The log-normal dispersion parameters (σ _ln D, Table 7) imply uncertainty bands of ± 10–15% in exceedance probabilities. This highlights the importance of interpreting the fragility curves probabilistically rather than deterministically.

• Damage state thresholds. The adopted MDR thresholds (0.1, 0.2, 0.5, 0.85) influence curve geometry. Sensitivity tests showed that shifting these thresholds by ±0.05 changes median fragility intensities by less than 0.2 MMI units, suggesting relative robustness but still warranting caution.

Taken together, these factors indicate that while absolute exceedance probabilities are subject to non-negligible uncertainty, the relative ranking of vulnerability classes and the overall conservative character of Algerian fragility curves remain robust.

Future research will seek to address these uncertainties within a fully probabilistic framework. Monte Carlo simulations, combined with explicit treatment of parameter distributions, will enable the propagation of both epistemic and aleatory uncertainties into fragility outcomes. Furthermore, integrating socio-economic exposure data will facilitate the construction of loss curves, providing decision-makers with confidence intervals not only on physical damage but also on economic consequences.

Finally, these uncertainty considerations should be read in conjunction with the typological limitations identified in Section 3.4. Together, they define the boundaries of applicability of the present vulnerability and fragility functions and highlight priorities for methodological refinement. Section 3.6 therefore turns to a broader discussion of the implications of these findings for seismic risk management in Algiers.

Sensitivity analysis

3.6

To identify which components of the VI most strongly influence the final vulnerability classification and the derived fragility/vulnerability functions, a sensitivity analysis was carried out using two complementary approaches.

1. One-at-a-time (OAT) perturbation. Each of the 12 VI parameters (Table 1) was systematically varied across its class levels (C1 → C2 → C3 → C4) while holding the other parameters constant at their baseline values. For each perturbation, the change in the building’s VI was computed and then propagated through the analytical MDR relationship (equation (2)) to evaluate its effect on the mean damage ratio at representative intensities (MMI = VII, VIII, IX).

2. Correlation-based ranking. Empirical correlation coefficients between the partial indices of individual parameters and the total VI (Table 2), calibrated against post-earthquake observations (Ain Temouchent 1999, Boumerdes 2003), were used to rank the parameters by relative importance. This data-driven measure of influence complements the OAT results and ensures methodological robustness.

Results

3.7

Both approaches produced consistent rankings of parameter influence. The most decisive contributors to vulnerability were:

• Total shear resistance of walls – the single most influential parameter, with the highest correlation values (Table 2). Shifting even one quality class in wall resistance leads to the largest changes in VI and, consequently, in MDR.

• General maintenance conditions – the second most influential; poor or absent maintenance significantly increases MDR across all intensities.

• Wall connections (tie/diaphragm quality) – ranked third; weak or absent connections raise deformation susceptibility and damage levels.

• Soil conditions – ranked fourth; soft sediments amplify shaking and elevate damage probabilities.

Parameters such as plan and elevation regularity and wall type exert a more moderate influence, individually less decisive but collectively relevant to overall vulnerability.