Optimized Design of Composite Cellular Beams: A Multi-Objective, Game- Theoretic, Environmental and Structural Assessment

Technical Summary: Optimized Design of Composite Cellular Beams

Problem Statement
The study addresses the design optimization of composite cellular beams integrated with steel deck slabs. While cellular beams offer high structural efficiency and reduced self-weight, their integration with concrete slabs introduces complex interactions requiring rigorous evaluation of mechanical behavior, failure modes, and environmental impacts. The core challenge lies in balancing three often conflicting objectives: minimizing construction costs, minimizing CO₂ emissions, and maximizing the slab's load-bearing capacity. Traditional design approaches often focus on single objectives (e.g., cost or structural capacity), potentially neglecting the trade-offs necessary for sustainable and efficient structural systems.

Methodology
The authors propose a multi-objective optimization framework employing two distinct strategies to solve the design problem:

1. Multi-Objective Particle Swarm Optimization (MOPSO):

   • Based on the foraging behavior of bird flocks, the MOPSO algorithm was extended to handle multiple conflicting objectives simultaneously.
   • The algorithm generates a Pareto front, representing a set of non-dominated solutions where improving one objective (e.g., reducing cost) would worsen another (e.g., increasing emissions or reducing load capacity).
   • The initial population was generated using Latin Hypercube Sampling to ensure diversity.

2. Game Theory Approach:

   • This method treats each objective function as a "player" in a cooperative game seeking a Nash Equilibrium.
   • The multi-objective problem is transformed into a single-objective problem by minimizing the product of relative losses (the deviation of each objective from its individual best value).
   • A normalization process is applied to improve convergence, and weights for the objectives are determined using the Entropy Method to avoid subjective decision-maker bias.

Design Variables and Constraints
The optimization considers a comprehensive set of design variables (Table 1 in the paper), including:

• Beam Parameters: Steel profile selection (laminated or welded), steel yield strength, circular opening diameter, spacing between openings, and the expansion factor of the beam section.
• Slab Parameters: Concrete thickness, steel deck thickness, concrete compressive strength, and reinforcement ratio.
• Constraints: The design must satisfy Brazilian Standards (NBR 8800:2024) regarding bending moment, vertical shear, and longitudinal shear capacities, as well as failure modes specific to cellular beams (e.g., Vierendeel bending).

Key Contributions and Results
The study evaluates the proposed frameworks across five distinct beam spans (10 m to 20 m) and compares the results against benchmark literature (Brandão et al.) and between the two optimization strategies.

• Pareto Front Generation: The MOPSO algorithm successfully generated diverse Pareto-optimal solutions, effectively illustrating the trade-offs between economic, environmental, and structural performance. The results showed robustness when compared to previous mono-objective studies, with solutions exhibiting proportional relationships in load capacity and CO₂ emissions.
• Game Theory Performance: The Game Theory approach consistently identified "compromise" solutions that balanced the three objectives. While these solutions were not strictly Pareto-optimal (as they arise from a scalarized formulation), they were found to be "near-Pareto optimal," lying in close proximity to the Pareto front.
• Solution Ranking: Using the Entropy Method to determine objective weights and Minkowski distance metrics (Manhattan, Euclidean, and Chebyshev) for ranking, the study found that Game Theory solutions ranked among the top alternatives. Specifically, for the shorter spans analyzed (10 m and 12.5 m), these solutions consistently ranked first or second. For longer spans (15 m, 17.5 m, and 20 m), they ranked fourth, fifth, and sixth, respectively.
• Design Trends:
  • Material Selection: Both approaches favored higher-strength steel grades for the beam profiles. For spans of 15 m or greater, welded sections were predominantly selected over rolled sections.
  • Concrete Strength: Solutions converged toward a concrete compressive strength of 25 MPa, though MOPSO solutions showed variability up to 50 MPa.
  • Efficiency vs. Capacity: Designs driven primarily by maximizing structural load-bearing capacity were consistently ranked as the least efficient alternatives, corresponding to significantly higher costs and CO₂ emissions.

Significance and Claims
The paper claims that the proposed framework demonstrates the potential of combining multi-objective optimization, Game Theory, and multi-criteria decision-making (via the Entropy Method) to support the sustainable and structurally efficient design of composite cellular beam systems.

The authors emphasize that while MOPSO provides a comprehensive view of the design space through the Pareto front, the Game Theory approach offers a practical, automated strategy for identifying balanced solutions without requiring subjective weighting from a decision-maker. The study concludes that seeking solutions that balance competing objectives is an effective strategy, particularly for shorter spans, and that designs focused solely on maximizing load capacity are generally inefficient from an economic and environmental standpoint. The authors modestly suggest that future work could extend these methodologies to other hole geometries, such as castellated beams or steel angelina beams, to verify the behavior of the solutions across different topologies.