Pedestrian Truss Bridge Optimization
Please download the image of the Grasshopper script below for a better view. It was difficult to fit the full script clearly within a single viewport.
Step 1: Generative Design Framework
- Design Decision 1: Pedestrian Truss Bridge Geometry and Sizing
- Design Variables
- Truss depth (height between top and bottom chords)
- Number of panels (controls panel length and diagonal spacing)
- Member cross-section diameter
- Top chord arch rise (parabolic curvature added to the top chord)
- Evaluators
- Total structural mass
- Represents material cost / embodied carbon / environmental impact
- Maximum vertical deflection under live load
- Represents serviceability
- First natural frequency
- Represents pedestrian-induced vibration comfort
- Maximum Euler buckling utilization across compression members
- Represents structural safety
- Most Important Tradeoffs to Consider
- Material cost vs. structural performance
- Larger, deeper, thicker designs will perform better structurally but use more steel and are assumed to be worse for the environment
- Vibration comfort vs. weight
- Lighter designs drop into the 2-5 Hz pedestrian resonance range and require added mass or active damping (additional cost for the developer)
- Member efficiency vs. safety
- Pushing buckling utilization toward 1.0 use material efficiently but eliminates safety reserves against buckling if live load is increased unexpectedly
- Design Decision 2: Long-Span Steel Arch Bridge
- Design Variables
- Arch rise (vertical height from springing to crown)
- Arch curve shape (parabolic vs. circular vs. catenary)
- Number of spandrel panels (columns from arch to deck)
- Arch truss depth
- Evaluators
- Total steel volume
- Horizontal thrust at abutments
- Increased thrust will equal larger foundation costs
- Maximum deck deflection under asymmetric live load
- Maximum compression force in any arch member
- Most Important Tradeoffs to Consider
- Arch rise vs. horizontal thrust
- Flat arches will transmit huge horizontal forces into the abutments —> Massive foundations
- Tall arches will use more material
- Structural elegance vs. constructibility
- Deep trussed arch carries asymmetric loads efficiently but requires more fabrication of complex connections
- Aesthetic prominence vs. functional minimalism
- Rise and depth choices balance visual impact against material economy
This was inspired by my trip to the Victoria Falls Bridge in Zimbabwe last summer (above).
- Design Decision 3: Through-Truss Pedestrian Bridge Enclosure Configuration
- Design Variables
- Truss depth above deck
- How far the side trusses extend above pedestrian eye level
- Diagonal pattern density
- Number of diagonal members per unit length
- Top lateral bracing presence
- Cross-bracing in top chords, Y/N?
- Member section transparency
- Slender vs. heavier sections
- Evaluators
- Total steel volume
- Average unobstructed view angle from a pedestrians eye height
- Lateral stability
- Maximum deflection
- Most Important Tradeoffs
- Structural efficiency vs. pedestrian view quality
- Denser triangulation improves stiffness but obstructs sightlines
- Open feel vs. lateral stability
- Removing top chord bracing creates a more open pedestrian experience at the expense of lateral stability
- Member slenderness vs. visual presence
- Thin members feel light and elegant but could buckle under compression
Step 2: Generative Design Study
For this assignment, I chose to develop Design Decision 1 (pedestrian truss bridge geometry and sizing) into a full generative design study.
Objective:
Identify the combination of truss depth, panel count, member diameter, and top chord arch rise that produces an optimal 40 meter simply-supported pedestrian truss bridge by balancing material economy (mass), serviceability (deflection), pedestrian vibration comfort (first natural frequency), and structural safety (compression member buckling utilization).
Model:
The model was developed in Grasshopper, with Karamba3D used to perform the structural analysis. The geometry generation workflow creates a Warren truss configuration consisting of parallel top and bottom chords with alternating diagonal members that form triangular panels. When a nonzero arch rise is specified, the top chord follows a parabolic profile, allowing the bridge geometry to transition between a conventional truss and an arch-supported system.
Rather than deriving member forces and deflections through the matrix structural analysis procedures typically implemented in advanced structural analysis software, the pedestrian bridge was analyzed directly in Karamba3D using a linear elastic finite element analysis (FEA). An eigenmode analysis was then performed to determine the first natural frequency of the structure. In addition, the Disassemble Cross Section component was used to extract cross-sectional properties, including the moment of inertia and modulus of elasticity, which were subsequently used to compute Euler critical buckling loads for the compression members.
Galapagos was used as the optimization engine, implementing a weighted-sum formulation to convert the multi-objective problem into a single-objective fitness function. Weight sweeps were performed across ten different objective prioritizations to generate a Pareto frontier illustrating the tradeoffs between competing design objectives.
- Design Variables
- Truss Depth
- From 2 m - 6 m, integer
- Number of Panels
- 6 to 16, integer
- Member cross-section diameter
- 10.0 cm - 25.0 cm, continuous
- Top chord arch rise
- 0 - 2 m, continuous
- Constants
- Span: 40.0 m
- Material: A500 Grade B structural steel
- E = 200 GPa
- Density = 7,850 kg/m^3
- Cross-section family: circular hollow section
- CHS/ O-Section
- Diameter-to-thickness ratio
- Fixed at 20 —> Wall thickness scales linearly with diameter
- Live load: 15 kN/m distributed along the bottom chord
- Simulates 5 kPa pedestrian load over a 3m deck width
- Boundary conditions: pin support at one end, roller at the other end, out-of-plane restrains along both chords
- Element type: beam elements (axial + bending) with linear elastic behavior
- Evaluators
- Total mass [kg]: Minimize
- Extracted from Karamba’s “Assemble Model Mass” output
- Maximum vertical deflection [cm]: Minimize
- Extracted by taking the max absolute value of nodal Z-displacements from the Karamba analysis output
- First natural frequency [Hz]: Maximize
- Extracted from Karamba’s “Eigen Modes” component, filtered to the first non-rigid-body mode
- Maximum Euler Utilization: Minimize
- Computed per compression member as F/ (pi^2 * E * I / L^2). Tension members are excluded by filtering Karamba’s “Beam Forces” output on sign before division
Optimization Setup
Rather than producing a Pareto front in a single optimization run (which would require a true multi-objective solver like Octopus which does not run reliably on a Mac), the tradeoff curve was traced manually by varying the weight coefficients of a single-objective fitness function across runs. Each run produced one optimal design, and the collected designs trace out the Pareto tradeoff.
Before any optimization with real weights, a pilot Galapagos run was conducted using arbitrary weights (m*0.0001 + d*10 + (1/f) + u*5) to exercise the design space broadly. Ten designs were sampled from the resulting population, spanning the full fitness gradient from best (1.97 to worst 26.01).
The observed minimum and maximum bounds for mass, max deflection, first frequency, and max utilization became the denominators for min-max normalization in the main fitness function.
From there, the following normalization function was constructed:
fitness = w1 * (m - 1500)/11500 + w2*(d - 0.02)/0.38 + w3*(1 - (f-1.5)/7.0) + w4*(u-0.05)/4.95
** Note the frequency term is inverted so that higher frequencies (which are desirable) reduce fitness, since Galapagos will minimize this function. Each normalized term will land in approximately [0,1] across the realistic design space, so the weights w1, w2, w3, w4 directly express the relative importance of each evaluator.
Ten Galapagos runs were then conducted, each with a different combination of weights to bias the optimizer toward different priorities:
The 10 rows from the table above were then plotted in Excel with total structural mass [kg] on the x-axis and maximum vertical deflection [cm] on the y-axis. A polynomial trendline was added to highlight this tradeoff. Each point was labeled with its run theme, making the relationship between priority structure and design outcome immediately visible. Three of the runs (Frequency-Priority, Utilization-Priority, and Performance-Balanced) converged to the same maximum-everything design and are represented by a single overlapping point at the lower right of the curve.
There are three insights that become evident when analyzing the curve above:
- The curve has clear regions with different cost-benefit ratios
- Below approx. 6,000 kg, small additions of material buy dramatic improvements in stiffness. Above 10,000 kg, deflection improvements become marginal.
- The lightest designs hit hard structural limits that aren’t visible on this chart
- The mass-priority and strong mass priority designs achieved maximum Euler utilization of approximately 1.0. This means that the compression members are at the buckling threshold with no margin of safety. The mass-priority design also has a first natural frequency of just 2.25 Hz, which falls within the 1.5-2.5 Hz range that infamously caused the London Millennium Bridge to have significant vibration issues
- There is no single optimal design, only optimal designs given a priority
- Each design is structurally valid but have different commitments between cost and performance.
If I were the designer of this pedestrian bridge, I would eliminate all designs below approx. 5,000 kg since they will struggle with frequency issues. In addition, they sit at the buckling thresholds.
Designs above approx. 11,000 kg are wasteful with additional steel producing negligible performance improvement.
I would advocate for a design between these extremes which would ultimately depend on how the client values material costs and performance (deflection and frequency). Regardless, this study would allow you to return to the client with better data instead of simply guessing on the exact tradeoffs.