Reliability Assessment of FRP-RC Columns Using AK-MCS with a Piecewise Two-Branch Approach

Nomenclature

• The governing capacity is the minimum of the stability and material capacities, while the resistance and action effect define the limit-state quantities.The two branches are denoted by ĝ_mat and ĝ_stab; model-uncertainty factors Θ_R and Θ_E apply to resistance and action effect.Source: Nomenclature · pp. 1–2
• The reliability notation distinguishes Monte Carlo, FORM/MPP, and independent-error reliability indices, as well as confidence bounds and their width.β_MCS and β_MPP denote Monte Carlo and FORM/MPP indices; Δβ is the width between β_lower and β_upper.Source: Nomenclature · p. 2

Column geometry and materials

• The column variables include geometry, material properties, loading, and reinforcement measures, with five inputs used as Kriging features.The listed quantities include slenderness λ, concrete strength f_c, modulus E_c, and FRP reinforcement ratio ρ_FRP.Source: Column geometry and materials · p. 3

1.1 Related work and identified gap

• Existing reliability methods leave unresolved the combination of single-mode FE observations and a broad transition zone between competing modes.Up to about 93% of training points can lie within one predictive standard deviation of the mode boundary in critical configurations.Source: §1.1 · pp. 4–5
• A single nonlinear FE analysis returns only the capacity of the governing mode, and which mode is censored depends systematically on the sample.The mode may be material failure or second-order instability at medium to high axial loads.Source: §1 · pp. 3–4, §1.1 · p. 4

1.2 Contribution and outline

• The two-branch framework trains each Kriging model only on FE analyses where its mode governs, then combines predictions by their minimum.It extends FORM to the two-branch surface and uses shifted surfaces for local confidence bounds on β.Source: §1.1 · p. 5, §1.2 · p. 5
• The paper adds post-convergence FE points using leave-one-out checks and confidence-interval width, then uses FORM design points and importance sampling for small failure probabilities.The approach is demonstrated on a representative column example.Source: §1.2 · p. 5

2.1 Motivation

• A single GP can overshoot beyond a sharp mode-transition kink because standard covariance kernels assume smoothness, whereas separate mode fits avoid fitting that kink.The FRP-RC column instead has a shallow crossing and broad transition zone.Source: §2.1 · p. 6, Figure 1 · p. 6

2.2 Standard AK-MCS

• Standard AK-MCS iteratively evaluates the candidate minimizing U(x) = |ĝ(x)|/σ̂(x) and stops when the minimum U reaches its threshold.The method assumes a single continuous failure mechanism.Source: §2.2 · pp. 6–7

2.3 Two-branch surrogate formulation and parallel prediction

• The two independent branch GPs are trained on FE points labeled by governing mode, and the combined prediction and its uncertainty come from the governing branch.If a branch has fewer than two points at termination, it remains unfitted and the model reduces to one branch.Source: §2.3 · p. 7

2.4 Enhancements for data-scarce branches

• A branch’s win-share estimates how often it governs the candidate pool and determines whether its fit quality matters.A branch governing 0.3% of the pool can be poorly fitted without affecting failure probability, unlike one governing 50%.Source: §2.4 · p. 7

2.4.1 Extrapolation safeguard

• Branch predictions are restricted to an extended axis-aligned box around training points to suppress gross extrapolation, though unobserved corners can remain inside it.The described configurations did not require excluding a branch for large-box uncertainty effects.Source: §2.4.1 · p. 7

2.4.2 Minimum population and relevance threshold

• When a relevant governing branch has fewer than 10 points, U is forced to zero in its region until it gains points, but forcing stops if it adds none within 20 refits.The branch must exceed a 1% win-share to receive this minimum-population treatment.Source: §2.4.2 · p. 8

2.4.3 Targeted space-filling sampling

• For underpopulated branches, targeted space-filling selects points by distance from existing samples, with a closest-point variant for specified low-load cases.After minimum population, five additional space-filling points are added in those cases before U-criterion refinement resumes.Source: §2.4.3 · p. 8

2.4.4 Isotropic kernel fallback

• An isotropic kernel with three hyperparameters is an optional fallback because fitting seven anisotropic hyperparameters to only 3–5 points can spuriously shrink predictive uncertainty.This option was used only for low load ratios with the targeted-sampling variant, not for the Section 4 examples.Source: §2.4.4 · p. 8

2.4.5 Classifier-assisted win-share

• A GP classifier estimates each mode’s pool win probability from FE labels without the branch bounding-box restriction, avoiding the direct estimate’s bias against data-scarce branches.It only determines whether a data-scarce branch receives points; it does not enter the surrogate prediction.Source: §2.4.5 · pp. 8–9

2.5 Model-uncertainty-aware active learning

• The surrogate is trained with ΘR = ΘE = 1 because the FE model supplies resistance only, while both uncertainty factors still enter the U-criterion.The known action effect lets the resistance prediction be recovered from the limit-state prediction without changing its predictive standard deviation.Source: §2.5 · p. 9
• Sampling ΘR and ΘE across the candidate pool places FE evaluations near the failure boundary while accounting for model uncertainty.Without this step, the FORM design point lay at concrete strengths up to about 8 MPa below the training range, where the surrogate extrapolated.Source: §2.5 · p. 9

2.6 Post-convergence refinement

• A converged U-criterion may still leave a wide β confidence interval or fail to reveal miscalibration in a data-scarce branch.The follow-up refinements use empirical calibration and add points to the branch driving uncertainty.Source: §2.6 · p. 9
• If a branch has max |zj| ≥ 3 and governs at least 1% of the pool, three points nearest its worst residual are evaluated per round, for up to three rounds.The leave-one-out check removes each near-limit-state training point in turn and refits the branch; calibrated residuals should lie roughly within [−3, +3].Source: §2.6 · p. 9
• If no branch is flagged, confidence-width refinement adds four points per round in the region contributing most pessimistic-only failures, stopping at the target or preset limits.The study target was ∆β = 0.1, not a code requirement; the target was reached only in part of the parameter study because ∆β falls slowly near equally likely modes.Source: §2.6 · p. 9

2.7 Overall procedure

• The overall procedure repeats active learning if refinement lowers min U below its threshold, then evaluates Pf and β using both MCS and FORM on the same surrogate.Because early two-point branch fits can vary substantially with optimizer starts, stopping requires min U ≥ uthreshold for three consecutive iterations.Source: §2.7 · p. 10

3.1 Confidence interval from the Kriging prediction uncertainty

• The confidence bounds βlower and βupper are obtained by shifting pool-point predictions by ±κσ̂ and converting failure-probability estimates to reliability indices; ∆β is their width.The bounds assume simultaneous shifts across pool points, and are meaningful only when σ̂ is calibrated.Source: §3.1 · pp. 10–11
• ∆β reflects plausible surrogate prediction error rather than Monte Carlo sampling error, so the paper reports the two uncertainties separately.The bounds are one-sided 97.5% bounds, more conservative than the draft DAfStb guideline’s one-sided 95% level.Source: §3.1 · p. 11

3.2 Calibration check via leave-one-out cross-validation

• Leave-one-out calibration checks are especially important for sparse branches, since likelihood-based hyperparameter fitting does not guarantee calibrated σ̂.A branch is flagged when max |zj| ≥ 3 and it governs at least 1% of the pool; the 1% condition left a branch with max |zj| = 6.93 unflagged when its share was 0.28%.Source: §3.2 · p. 11

3.3 Correlated versus independent surrogate error and the use of the bounds

• Independent surrogate errors shifted β by 0.0044 and 0.0041 in the two example configurations, roughly 1/35 of confidence widths 0.151 and 0.153.The nominal β is treated as the best estimate, while a conservative lower bound is needed for verification against a target reliability; both conclusions require calibrated σ̂.Source: §3.3 · pp. 11–12

3.4 FORM/MPP search on the two-branch Kriging limit-state surface

• βMPP remains available when finite-pool MCS finds no failures; in l60n07 and l60n09 it exceeds βMCS by 0.05 and 0.04, respectively.An FE analysis at the design point confirmed the surrogate prediction.Source: §3.4 · p. 12

3.5 FORM on the shifted limit-state surfaces

• The shifted-surfaces interval ∆βFORM was smaller than ∆β in 69 of 73 configurations with Pf > 0 in the pool, with median ratio 0.52.It reflects uncertainty near the design point and does not replace ∆β; it provides an interval when the pool contains no failure.Source: §3.5 · p. 12

3.6 Importance sampling at the FORM design points

• Importance sampling mixes standard-normal densities centered on up to four nominal, pessimistic, and branch-specific design points, weighting samples to estimate Pf.Direct MCS requires about 1600/Pf samples to reach the cited 2.5% coefficient-of-variation criterion, exceeding 10^12 at β = 6.Source: §3.6 · pp. 12–13
• A fixed importance-sampling mixture can undersample a distant failure region and underestimate Pf, even when block-wise variation does not reveal the miss.Branch-specific centers address the risk of missing a mode when the combined design point lies on the stability side but material failure contributes many pool failures.Source: §3.6 · p. 13

4.1 Column configuration

• Omega-04 is a slender, simply supported FRP-RC column; the examples include mixed-failure l60n07 and l60n09 and stability-dominated l80n04.All considered slenderness ratios exceed the proposed short-column limit by at least a factor of 1.8.Source: §4.1 · pp. 13–14, Figure 3 · p. 13

4.2 Finite-element model and failure-mode classification

• The FE model classifies failure by whether a specified strain limit is reached before peak load; otherwise, peak load is classified as stability failure.Arc-length continuation traces the equilibrium path, and the material branch combines concrete crushing and FRP rupture; the example configurations exclude cases where FRP rupture governs.Source: §4.2 · p. 16

4.3 Results

• The final l60n07 and l60n09 training sets contain 71 and 56 points, and their final ∆β values of 0.128 and 0.147 exceed the 0.1 target.Convergence held after follow-up, and all leave-one-out residuals remained below 3.Source: §4.3 · pp. 16–17
• For l60n07 and l60n09, βMPP exceeds βMCS by 0.047 and 0.036 (1.1% and 0.9%); for l80n04, βIS = 6.187 despite no direct-pool failures.At the combined design point, FE resistance and the Kriging prediction differ by at most 0.58%.Source: Table 4 · p. 17, §4.3 · p. 17, Table 5 · p. 17
• In l60n07 and l60n09, 79% and 62% of training points lie within one predictive standard deviation of the mode boundary.Across 60 configurations with both modes, the median share is 65% and the maximum is 93%; FE-analysis shares also differ from each mode’s share of pool failures.Source: §4.3 · pp. 17–18

4.4 Single-branch fallback and an irrelevant, miscalibrated branch

• When all 38 l40n04 analyses produce material failure, the surrogate uses the material branch alone without further intervention.Formal verification for that case would still require importance sampling.Source: §4.4 · p. 18
• In l40n07, the poorly calibrated stability branch governs only 0.28% of the pool; refining it would not change the 158 pool failures or β = 4.162.None of the pool failures is governed by that branch.Source: §4.4 · p. 18

5 Discussion

• The two-branch formulation is limited to two modes; at ν = 0.1, concrete crushing and FRP rupture compete, warranting a third branch test.More branches receive fewer points, making data starvation more severe; bounding boxes also leave empty corners where extrapolation can occur.Source: §5 · pp. 18–19
• The authors leave open whether one Kriging surrogate without branch-splitting would yield the same reliability index and local calibration.They also note that the fixed importance-sampling density cannot detect failure regions far from its centers; adaptive densities are proposed as a possible alternative.Source: §5 · p. 19
• A median of 63 FE analyses (range 17 to 172) were required across 84 configurations, with 35% from post-convergence refinement.The authors note this cost is uncritical at about one second per analysis but intend the method for much longer FE runs.Source: §5 · p. 19

6 Conclusions

• 79% and 62% of training points in mixed-failure configurations lie within one predictive standard deviation of the mode boundary; design-point FORM agrees with Monte Carlo within 0.05.An FE analysis at the design point confirmed the surrogate to within 0.6%.Source: §6 · p. 19
• Data-scarcity safeguards keep the method operational when one mode is rare or absent, while a branch governing less than 1% of the candidate pool is not refined.Post-convergence refinement based on leave-one-out checks and confidence-interval width controls the accuracy of β and βlower; U-criterion convergence alone is insufficient.Source: §6 · p. 19
• Importance sampling at design points gave β = 6.19 with a coefficient of variation of Pf of 0.8% from 2.5 · 10^5 samples, while direct MCS with 10^7 samples found no failure.FORM on shifted limit-state surfaces gave a local confidence interval narrower than ∆β in 69 of 73 configurations.Source: §6 · p. 19
• Whether the findings generalize across the full design space and whether the method can replace FE-based Level-III analysis under D375 and EN 1990 remain under study.Source: §6 · p. 19

• Goldack is credited across the study’s conceptualization, methods, software, analysis, writing, and supervision; Koehler and Syed contributed resources, validation, and review and editing.Source: CRediT authorship contribution statement · p. 19

• The research received no specific grant from public, commercial, or not-for-profit funding agencies.Source: Funding · p. 20