Frame design with a surrogate model

A Latin hypercube picks N frame designs; each is analysed with a linear 3D frame model: 12-degree-of-freedom beam elements, rigid floors, fixed bases, the EN 1990 combinations 6.10 with the recommended factors for wind on either face, sway imperfections (EN 1993-1-1, 5.3.2), Horne’s $\alpha_{\mathrm{cr}}$ with sway amplification (5.2.2) and simplified member screens.

The earthquake follows the lateral force method of EN 1998-1 (4.3.3.2): the period $T_1$ from Rayleigh’s quotient with the floor masses of $G + \psi_{E}Q$, the type 1 design spectrum with $q = 1.5$, the base shear $F_{b} = S_{d}(T_1)\,m\,\lambda$ spread over the floors in proportion to height and mass, an accidental torsion of 5 % of the plan size, and the combinations $G + \psi_{2}Q + E$ with $E = E_x + 0.3E_y$ or $0.3E_x + E_y$. Each direction’s effects are amplified by $1/(1-\theta)$ where $\theta > 0.1$ (4.4.2.2). The earthquake check is the larger of the damage-limitation drift $\nu d_r / 0.0075h$ at the plan corners (4.4.3.2, $\nu = 0.5$) and $\theta / 0.2$; the members are screened under the seismic combinations too.

A Gaussian process per output (drift, utilization of three member groups, $\alpha_{\mathrm{cr}}$, the earthquake check; Matérn 5/2 with one length scale per variable) is fitted to these runs by maximizing the marginal likelihood. Its accuracy is measured by leave-one-out with the same hyperparameters.

The map shows what the surrogate has learned: for every design at one upper-column size, whether it predicts all limits met with its $\pm 2\sigma$ band inside them, unsure (the band crosses a limit) or not met, and the predicted limit where $\mu$ crosses it. Each FE run you add joins the training set and the surrogate is refitted (infill; Forrester, Sóbester and Keane 2008).

A genetic algorithm then minimizes steel mass on the surrogate, keeping $k$ standard deviations away from every limit (designs already analysed count with their FE result), and the frame model checks the answer; a failed check joins the training set and the search runs again (surrogate-assisted evolutionary design, Jin 2011). Reveal analyses all 1,296 catalog designs as the honest baseline and grades the surrogate against it.

This is the workflow I use in my research: surrogates trained on expensive analyses, searched by an optimizer and checked by the full model.

A teaching model, not a design tool: wind on either face with a simplified net pressure coefficient of 1.3 and no wind torsion, rigid floors, fixed bases, no connection design, no checks of the floor beams’ lateral stability (the slab is assumed to hold them), S355 with $\gamma_{\mathrm{M0}}=\gamma_{\mathrm{M1}}=1.0$.

The column check $N/(\chi_{\mathrm{min}}N_{\mathrm{pl}})+M_{y}/M_{\mathrm{pl},y}+M_{z}/M_{\mathrm{pl},z}$ with a storey-height buckling length is a simplified screen, not the EN 1993-1-1 interaction formula. Beams are checked for bending and shear separately. $\alpha_{\mathrm{cr}}$ comes from Horne’s storey method in both directions; amplification applies to the sway effects only.

The map is a slice at one upper-column size; FE runs at other sizes are drawn faint, and the GA moves in all three variables. The run budget and the hour per analysis are a thought experiment: here one analysis takes milliseconds.

The drift limits $h/300$, $h/400$ and $h/500$ are illustrative screening limits. The $\pm 2\sigma$ bands are model-based intervals of the surrogate, not probabilities of meeting a code; $k=2$ is not a 98 % reliability. The exhaustive optimum is exact only within this catalog and this simplified model.

The earthquake design is ductility class low ($q = 1.5$) with the member screens above and no capacity design. EN 1998-1 recommends that class only for low seismicity (6.1.2); at $a_{g} = 0.15\,g$ a real frame would normally be designed for ductility class medium with capacity design, so read this as a screening check. The lateral force method needs $T_1 \le \min(4T_{C}, 2\,\mathrm{s})$ and a building regular in elevation; the page says when a design’s period is beyond that. $\theta$ above 0.2 counts as failing, as it would need a second-order analysis. $\varphi = 0.8$ for every storey in $\psi_{E}$; importance class II; ductile non-structural elements.

Loads are stated assumptions you can change. Hand check before publication: the default design against a commercial frame program.