Nathan Cao

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Step 1 - Generative Design Framework

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Step 2 - Generative Design Study

The study focused on Design Decision 1: column grid bay spacing. The goal was to understand how the choice of bay configuration on a fixed-sized floor plate drives the trade-off between framing cost and column-free space. The optimization was done in Grasshopper using Wallacei X, which is a component that is able to take multiple objectives and optimize for multiple evaluators (this was done because Octopus, which Notion suggests using, is no longer compatible with most Rhino versions). For this assignment, a 40’ by 40’ floor plate was studied, although different aspect ratios are discussed at the end of this entry.

Following the framework:

Design Variables (2 for 2 units)

Num_Bays_X and Num_Bays_Y, each ranging from 1 to ~10 bays. Bay dimensions are computed as Total_Width / Num_Bays_X and Total_Length / Num_Bays_Y, then converted from feet to millimeters for the weight formulas.

Evaluators (Wallacei Objectives)

Minimize Total Beam Weight — computed as the sum of beam_weight_x, beam_weight_y, and column_weight expressions. The beam weight formulas (x² · 0.5 · y · (z+1)) capture the quadratic relationship between span and required steel, multiplied by tributary width and perpendicular bay count. Column weight scales linearly with total column count via (x+1)(y+1)·3.
Maximize Column-Free Area — bay_width × bay_length, multiplied by −1 since Wallacei minimizes by default (finds the most negative solutions, which end up maximizing bays).

Wallacei Settings

Population and generation counts as configured (visible in the Wallacei X component). To run the analysis, click on the Wallacei X component once the inputs are selected. Outputs were analyzed using Wallacei's Parallel Coordinate Plot and Fitness Values Objective Space (scatterplot) components.

Simplifications and assumptions for this optimization problem

In reality, optimizing bays is a bit more nuanced than what is being implemented, but the general gist is there. The following assumptions/simplifications were made:

Floor plate is fixed rectangular grid. Real projects often have irregular footprints, setbacks, cores, and program-driven constraints that force non-uniform bay layouts. This study assumes a clean rectangular plate.
Beam weight is approximated by a closed-form expression rather than actual member design. The formula (span² · 0.5 · tributary · (perpendicular bay count + 1)) captures the quadratic relationship between span and required steel (based on moment demand = wl^2/8), but it does not select real W-shapes from the AISC database, check flexural or shear capacity, or verify deflection limits (L/360, L/240). A real design would size each beam for ASCE 7 gravity loads, considering composite or non-composite action.
Column weight scales linearly with column count via a constant multiplier. In reality, column size is driven by accumulated tributary area, number of stories, and lateral demands. A 2-bay layout has fewer columns, but each carries more load and would need a heavier section; the current formula doesn't capture this load-driven sizing.
Two-way slab framing was assumed. Assuming a two-way slab load path (like a waffle slab) allows all beams to have the same assumed section, simplifying the analysis. A one-way slab system with infill beams could be more structurally efficient, but would add complexity to the optimization.

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Step 3 - Generative Design Study Results

Parallel Coordinate Plot of All Solutions from Gene Pool For a Square Floor Plate

Scatter Plot (With Pareto Front) for a Square Floor Plate

The scatterplot shows the objective space from Wallacei, with column-free area on the X-axis (plotted such that "right" corresponds to larger bays, since the raw value is negative) and framing cost on the Y-axis (plotted such that "up" corresponds to lower, more cost-effective solutions).
From the scatterplot, the trade-off between the two evaluators is clearly visible as a well-defined Pareto front. The non-dominated solutions identified by Wallacei form a continuous front, confirming that framing cost and column-free bay size are genuinely competing objectives. No configuration achieves both low cost and large bays simultaneously. The density of points along the front is uneven: solutions cluster more heavily toward the upper-left region (lower framing cost and framing weights, smaller bays), while the large-bay, high-cost end of the front is sparser.
This plot turns a single-answer problem into a menu of trade-offs. Rather than picking a bay layout based on a rule-of-thumb (e.g., 30 ft typical), I can see exactly what each configuration costs and what each delivers in terms of column-free area (relative to each other, as Wallacei's scatter plot does not display raw evaluator values, but the Parallel Coordinate Plot does, allowing direct comparison of framing cost and bay dimension across solutions). Due to how linear the Pareto Front is, the trade-off rate between framing cost and bay openness is constant across the entire front, meaning the final choice comes down to how much the project values column-free space versus cost efficiency. I would likely choose a solution away from the cluster in the upper left corner (slightly higher framing cost for more bay openness), as this shows configurations that still benefit from a lot of openness, but before bay size starts to shrink significantly. This behavior is also evident in the PCP, where you can see a cluster of lines that emphasize minimizing framing cost at the expense of column-free area. This is “the best bang for your buck”/“best of both worlds”.

Additional Notes

To turn on or off the different plots (PCP or scatterplot), turn the preview on or off.
For extra credit and better visualization, I also included a rendering of the selected bay configuration (to turn it on or off, just turn on or off the merge at the end of the rendering group).
Wallacei automatically minimizes all objectives, so the column-free space objective was entered as a negative value to convert the maximization problem into an equivalent minimization (meaning more negative values (toward -121,920) represent larger, more desirable bays).
The write-up in this journal entry refers to a square floor plate configuration. For a higher aspect ratio, the behaviour is similar, but the scatter plot is more dispersed and even a little bit curved (see below).

Scatterplot for a floor plate with a 2:1 ratio (with noticeably more dispersion of solutions and less linearity). Asymmetry between X and Y directions creates a non-uniform trade-off rate, pulling solutions apart and introducing more curvature into the front.

Wiring