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A Strut and Tie Neural Network Surrogate for Failure-State Prediction of Concrete D-Regions Designed by the Compatible Stress Field Method

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DOI:

https://doi.org/10.31224/7250

Keywords:

Compatible Stress Field Method, strut and tie model, discontinuity regions, neural-network surrogate, machine learning, surrogate model, structural concrete, failure load

Abstract

The Compatible Stress Field Method (CSFM) designs the discontinuity regions (D-regions) of structural concrete, such as deep beams, hammerhead piers and pile caps, by combining a lower-bound stress field with kinematic compatibility and realistic constitutive behaviour, but its nonlinear solution procedure is too slow for the repeated evaluations that design-space exploration, reinforcement optimisation and reliability analysis demand. This study presents a neural-network surrogate that, in a single forward pass, maps a normalised description of a D-region design to its failure load factor and to the strut and tie (STM) member forces at the failure state, so the surrogate returns an interpretable and nearly statically admissible force state rather than an opaque capacity number. Training data come from sweeping an existing reference solver over a Latin Hypercube design of experiments across four D-region archetypes: deep beam, hammerhead pier, multi-column bent and pile cap. Trained per archetype, the surrogate predicts the failure load factor with a coefficient of determination of 0.96-0.99 and a mean absolute percentage error of 3-6 %, ahead of a panel of seven tabular baselines, and reproduces the reference member forces to within about 6 % in relative error (coefficient of determination above 0.99), with a nodal-equilibrium residual of about 1.5 % of the applied load. A bagged deep ensemble with split-conformal calibration attaches a prediction interval with a finite-sample coverage guarantee to each output, and a domain-of-validity flag built on the ensemble spread catches the majority of out-of-domain designs, on which accuracy degrades as expected. The reference solver is itself checked against an experimental pier-cap benchmark, completing a two-tier validation, though that tier rests on a single five-specimen series and inherits a systematic conservative bias. Against the reference solver the surrogate is between one and nearly four orders of magnitude faster, depending on the archetype and on batching, which makes the design-space exploration, reinforcement optimisation and reliability analysis of D-regions tractable.

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Posted

2026-06-03 — Updated on 2026-06-22

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Copyright (c) 2026 Sandesh Lamsal, Rubi Bhandari

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Version justification

This manuscript is a substantial revision and extension of the earlier draft. The main changes: 1. Force-state output (corrected, now central). The member-force prediction head was previously mis-scaled and therefore uninformative; it now predicts forces normalised by the applied-load magnitude, so the surrogate returns an interpretable, nearly statically admissible strut-and-tie force state (member-force coefficient of determination above 0.99; nodal-equilibrium residual about 1.5 % of the applied load) at no cost to failure-load accuracy. The title was changed from "Failure-Load" to "Failure-State Prediction" to reflect this co-equal contribution. 2. Seven new experiments: a seven-model tabular-baseline panel (ridge, polynomial surface, k-nearest neighbours, support-vector regression, random forest, gradient-boosted trees, Gaussian process); out-of-domain extrapolation with a domain-of-validity flag; member-force and nodal-equilibrium statistics; a force-loss / equilibrium-weight ablation; a non-failing (load-capped) design classification with no false negatives; learning curves with five-split stability; and per-archetype split-conformal calibration. 3. Calibrated uncertainty: a bagged deep ensemble with split-conformal calibration gives each prediction an interval with a finite-sample coverage guarantee, plus a domain-of-validity flag that catches the majority of out-of-domain designs. 4. Honest, two-tier validation: the surrogate is checked against the reference solver and the solver against an experimental pier-cap benchmark, with an explicit acknowledgement that this rests on a single five-specimen series with a systematic conservative bias, and the speed-up reported as an honest range (one to nearly four orders of magnitude) rather than a single figure. 5. Clearer framing and positioning: the discrete reference solver is kept distinct from the continuum CSFM throughout; the work is cast as an assessment task (given the geometry and reinforcement, predict the load-bearing capacity); and it is positioned against the recent reverse-engineering of strut-and-tie models by Yu and Kaufmann (Structural Concrete, 2026). 6. Structure and writing: the literature review was integrated into a single Introduction (no separate "Related Work" section); the Conclusions were rewritten as flowing prose; a Nomenclature section was added; the abstract now reports both the relative member-force error and the coefficient of determination; and the "non-failing design" terminology was unified throughout the text, figures and tables.