Retaining Wall Study Shows Cost and Embodied Carbon Align Nearly Perfectly

In a rigorous investigation into structural sustainability and computational engineering, a research study published in Discover Civil Engineering has upended conventional assumptions regarding the design of reinforced concrete cantilever retaining walls. Authored by Pham Ngoc Thinh, the investigation established an advanced multi-objective, reliability-based design optimization framework that simultaneously evaluates construction expenses and embodied greenhouse gas emissions under strict probabilistic safety constraints. Contrary to widespread assumptions that economic efficiency must compete against environmental sustainability, the research demonstrated that construction cost and embodied carbon are exceptionally aligned across the entire feasible design space, exhibiting a near-perfect Pearson correlation coefficient of 0.9957. Furthermore, the inquiry revealed an unexpected computational reality regarding machine-learning surrogate models, which are routinely deployed to bypass heavy simulations: when applied to explicit algebraic limit states, surrogate models actually performed substantially slower than direct vectorized analytical calculations, while introducing tail errors that were three to ten times larger than Monte Carlo sampling noise.

Cantilever retaining walls represent ubiquitous infrastructure assets across civil engineering, highway construction, earthwork stabilization, and urban development, yet their geometric proportions and reinforcement layouts are historically determined through iterative trial sizing governed by deterministic factors of safety. Although this traditional engineering workflow successfully prevents structural collapse, it produces overly conservative structural cross sections that demand far more concrete and reinforcing steel than static equilibrium requires, hiding surplus material and carbon footprints from design practitioners. Over recent years, metaheuristic search algorithms, including genetic algorithms, particle swarm optimization, and differential evolution, have been deployed to automate geometry generation and minimize overall capital costs. However, earlier numerical formulations almost universally treated the surrounding ground as a uniform, deterministic medium, assigning fixed values to soil friction angle, internal cohesion, and unit weight. In natural geotechnical formations, soil properties vary substantially across space, exhibiting coefficients of variation ranging from 10 percent to 40 percent.

When material minimization strips away excess structural bulk, retaining walls lose the uncalculated reserves necessary to accommodate soil variability, causing genuine probabilities of sliding, overturning, or foundation bearing failure to escalate far beyond nominal safety expectations. Reliability-based design optimization rectifies this vulnerability by imposing explicit probabilistic constraints on limit-state failure probabilities. For the benchmark 6.0-meter cantilever wall retaining granular backfill under American Concrete Institute standards analyzed in this study, the target reliability index was set at 3.0, corresponding to an allowable failure probability of 1.35 times 10 to the negative third power. Enforcing such probabilistic criteria inside an iterative population search creates a nested double loop: evolving a population of 60 candidate designs across 250 evolutionary generations while executing 50,000 crude Monte Carlo evaluations per candidate yields 7.5 times 10 to the eighth power limit-state calculations per optimization run. Addressing this computational burden typically necessitates constructing machine-learning surrogate approximations.

To evaluate surrogate performance systematically, the investigation tested three distinct regression architectures against an identical fourteen-dimensional Latin hypercube design of experiments spanning eight design variables and six geotechnical random variables. The evaluated families comprised extreme gradient boosting through regression tree ensembles, ordinary Kriging equipped with an anisotropic Matern five-halves covariance kernel, and sparse polynomial chaos expansion fitted via least-angle regression. On standard regression metrics, ordinary Kriging achieved remarkable accuracy on 2,000 training points, yielding coefficients of determination between 0.9923 and 1.0000 across all four governing limit states: overturning, sliding, foundation bearing capacity, and stem flexure. Sparse polynomial chaos expansion demonstrated mathematical perfection on the three smooth limit states, securing coefficients between 0.9975 and 0.9991 and outperforming the tree ensemble by an order of magnitude in residual error. However, polynomial chaos saturated abruptly at a coefficient of 0.9013 on the bearing capacity limit state, failing to improve even with expanded training datasets.

The performance bottleneck observed in polynomial chaos stemmed directly from underlying physical mechanics. Foundation bearing capacity depends on load eccentricity measured from the base center, introducing an absolute value that creates a derivative discontinuity at zero eccentricity, alongside a reciprocal pole corresponding to the boundary of triangular soil contact stress. While smooth global orthogonal polynomials cannot capture sharp kinks without severe numerical oscillations, the piecewise axis-aligned splits of extreme gradient boosting adapted effectively, attaining a coefficient of determination of 0.9544 at 10,000 training samples. More critically, the study established that high coefficient-of-determination scores on response values do not guarantee accuracy in reliability estimation. Propagating surrogate predictions into the minimum reliability index across 40 test designs near the feasible boundary revealed mean absolute errors of 0.106 for Kriging, 0.205 for gradient boosting, and 0.356 for polynomial chaos. These surrogate-induced errors dwarfed the baseline Monte Carlo sampling standard deviation of 0.037.

Compounding this discrepancy, both deployable surrogate families exhibited positive systematic bias, artificially overestimating structural reliability indices by 0.160 for gradient boosting and 0.306 for polynomial chaos. Consequently, the surrogates permitted unsafe designs to penetrate the feasible archive, misclassifying candidate geometries that violated code requirements. Conversely, ordinary Kriging maintained an unbiased mean error of negative 0.032, but its cubic training complexity and expensive matrix operations demanded 472 seconds per million evaluations, requiring approximately 4.1 days of continuous computation for a single optimization run. In sharp contrast, direct vectorized analytical evaluation of the closed-form limit states consumed merely 0.128 seconds per million evaluations, rendering the true mechanical formulation 21 times faster than polynomial chaos and 107 times faster than gradient boosting. A derived break-even criterion revealed that surrogate construction becomes computationally advantageous only when the true structural analysis takes longer than 2.8 microseconds per evaluation, proving counterproductive for closed-form retaining wall formulations.

Having established the operational superiority of direct analytical limit states, the study benchmarked multi-objective metaheuristic optimizers under an identical evaluation budget of 15,000 design evaluations across ten independent random seeds. The evaluated algorithms included a hybrid combining the Grey Wolf Optimizer with differential evolution, standalone Grey Wolf Optimizer, standalone differential evolution, and the non-dominated sorting genetic algorithm known as NSGA-II. Performance metrics demonstrated that the hybrid algorithm achieved an average hypervolume of 8.3975 against a fixed reference point, decisively outperforming standalone Grey Wolf optimization at 8.0045 and NSGA-II at 7.8725, with both comparisons yielding two-sided Mann-Whitney U test p-values of 0.00029. Both inferior algorithms suffered severe population diversity collapse, converging to barely one non-dominated solution per run. However, the hybrid proved statistically indistinguishable from standalone differential evolution, which attained a hypervolume of 8.4605 with a p-value of 0.18588, demonstrating that difference-vector mutation drives algorithmic search effectiveness.

At the recovered reliability-consistent optimum, the hybrid framework identified an efficient structural geometry requiring an expenditure of 873 United States dollars and generating 2,602 kilograms of carbon dioxide equivalent per linear meter of wall. At this optimal configuration, three limit states became active simultaneously: base sliding, foundation bearing capacity, and stem flexure all rested within 0.01 of the target reliability index on the search sample, while overturning stability maintained a massive safety reserve at an index of 4.61. Because concrete volume dominates both financial cost and carbon footprint, with concrete costing approximately 0.29 dollars per kilogram of embodied carbon compared to 0.48 dollars for reinforcing steel, reducing structural volume lowered both objectives concurrently. The recovered non-dominated Pareto front collapsed into a narrow band spanning 873 to 878 dollars per meter in cost and 2,588 to 2,602 kilograms of emissions, showing that cost-optimal and carbon-optimal geometries differ by only 0.5 percent.

While algorithmic refinements produced modest adjustments, the mathematical characterization of subsurface soil properties proved to be the single most powerful driver of optimal design. Incorporating realistic cross-correlation between soil cohesion and internal friction angle through a Nataf transform reduced the optimal construction cost by 25 percent to 653 dollars per meter, shifting the critical governing limit state from bearing capacity to base sliding. Applying Vanmarcke spatial averaging across mobilized failure surfaces to account for geotechnical variance reduction lowered the optimal cost by 28 percent to 627 dollars per meter. Crucially, testing sensitivity to soil uncertainty revealed a strict feasibility threshold: scaling baseline soil coefficients of variation upward by merely 10 percent caused the feasible design space to vanish entirely, as the maximum attainable reliability index dropped below the required threshold of 3.0. These geotechnical influences shifted design metrics by magnitudes up to five times greater than algorithmic operator variations.

Finally, the research diagnosed and resolved an inherent statistical vulnerability in simulation-based optimization known as search bias. When metaheuristic algorithms search candidate designs using common random numbers, the optimizer selectively exploits favorable stochastic fluctuations, artificially inflating the reported reliability index by an average of 0.047. Implementing an explicit safety buffer of 0.05 during the search successfully eradicated this bias on independent validation samples, though it introduced an 8.5 percent cost penalty, raising construction expenditure from 805 to 873 dollars per meter. The author outlined essential limitations, noting that closed-form equilibrium equations do not capture seismic acceleration, wall deflections, or numerical failure paths through heterogeneous soil deposits. Nevertheless, the study delivers clear guidance for structural and geotechnical practitioners: because embodied carbon closely mirrors financial expense in standard reinforced concrete, engineering resources are far better directed toward rigorous geotechnical site characterization and spatial variance modeling than toward complex multi-objective trade-off negotiations.

Subject of Research: Reliability-based design optimization of reinforced concrete cantilever retaining walls

Article Title: Optimizing reinforced concrete cantilever retaining walls for cost and embodied carbon under reliability constraints using surrogate models and hybrid metaheuristics

Article References: Thinh, P. N. (2026). Optimizing reinforced concrete cantilever retaining walls for cost and embodied carbon under reliability constraints using surrogate models and hybrid metaheuristics. Discover Civil Engineering, 3(1), Article 236. https://doi.org/10.1007/s44290-026-00649-x

Image Credits: AI Generated

DOI: 10.1007/s44290-026-00649-x

Keywords: retaining walls, reinforced concrete, reliability-based design optimization, embodied carbon, construction cost, surrogate models, Kriging, polynomial chaos expansion, extreme gradient boosting, Monte Carlo simulation, limit states, bearing capacity, differential evolution, Grey Wolf Optimizer, geotechnical uncertainty, spatial averaging, Nataf transform