Optimales Brückendesign mittels Ameisenkolonie-Optimierung (Optimizing structural design of bridge using ant colony optimization)

Overview

• Load Requirements:
  • Dead Load = 200 kN
  • Live Load = 800 kN
  • Dynamic Load = 300 kN
• Environmental Factors:
  • Wind Load = 100 kN
  • Seismic Factor = 0.5
• Cost Constraint: Max Budget = $5000
• Span Length: Between 20 m and 60 m
• Components: 10 structural elements

Objective

Minimize:
1. Total weight of the bridge
2. Construction complexity

Subject to:
1. Load capacity ≥ Total static load (Dead + Live)
2. Dynamic load capacity ≥ Required dynamic load
3. Cost ≤ $5000

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Solution Steps

• Define Material Choices

Material	Density (g/cm³)	Strength (MPa)	Cost ($/kg)	Fatigue Resistance	Corrosion Resistance
Steel	7.85	500	50	0.8	0.6
Aluminum	2.7	300	80	0.7	0.8
Concrete	2.4	40	20	0.9	0.5

• Determine Cross-Sectional Areas

Possible cross-sectional areas: A ∈ {10, 20, 30, 40, 50} cm²

• Choose Component Heights

Height range: h ∈ [1, 5] m

• Select Span Length

Discrete options: L ∈ {20, 30, 40, 50, 60} m

• Assign Critical vs. Non-Critical Components

  • Critical components (e.g., main beams) must support the primary loads.
  • Non-critical components (e.g., bracings) contribute to stability.

• Structural Analysis

  • Weight Calculation: W = Density × Length × A × 10^(-3) kg
  • Strength Check: Max Load Capacity = min(Strength×A) × (Redundancy Factor/Safety Factor), where Redundancy Factor = 1.2, Safety Factor = 1.5
  • Dynamic Load Check: Fatigue Limit = Fatigue Resistance × Strength
  • Environmental Stability: Env. Factor = 1 − (Wind Load/1000 + Seismic Factor/10); Adjusted Load Capacity = Max Load Capacity × Env. Factor

• Cost Calculation

  • Total Cost = ∑(Material Cost × Weight)

• Fitness Evaluation

  • If all constraints are satisfied: Fitness = Total Weight + (Construction Complexity × 100)
  • Else: Fitness = ∞ (invalid solution)

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Sample Calculation

• Span Length (L): 40 m
• Components:
  • 5 critical (steel, A = 30 cm², h = 3 m)
  • 5 non-critical (aluminum, A = 20 cm², h = 2 m)
• Component Length: Length per component = 40/5 = 8 m
• Weight:
  • Steel: 7.85 × 8 × 30 × 10^(-3) = 1.884 kg
  • Aluminum: 2.7 × 8 × 20 × 10^(-3) = 0.432 kg
  • Total Weight = 5 × 1.884 + 5 × 0.432 = 11.58 kg
• Strength Check:
  • Steel: 500 × 30 = 15000 kN
  • Max Load Capacity = (15000 × 1.2)/1.5 = 12000 kN
  • Required Static Load = 200 + 800 = 1000 kN ⟶ OK
• Dynamic Load Check:
  • Fatigue Limit = 0.8 × 15000 = 12000 kN
  • Required Dynamic Load = 300 kN ⟶ OK
• Environmental Stability:
  • Env. Factor = 1 - (100/1000 + 0.5/10) = 0.85
  • Adjusted Capacity = 12000 × 0.85 = 10200 kN > Wind Load (100 kN) ⟶ OK
• Cost:
  • Steel: 50 × 1.884 = 94.2$
  • Aluminum: 80 × 0.432 = 34.56$
  • Total Cost = 5 × 94.2 + 5 × 34.56 = 643.8$ < $5000 ⟶ OK
• Fitness:
  • Construction Complexity 10/50 = 0.2
  • Fitness = 11.58 + (0.2 × 100) = 31.58

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Optimal Solution

• After evaluating multiple designs, the best solution is selected based on:
  • Minimum Fitness Value (lowest weight + complexity).
  • Constraint Satisfaction (loads, cost).
• Solution
  • Span Length = 40 m
  • Materials = Steel (critical), Aluminum (non-critical)
  • Cross-Sectional Areas = 30 cm² (critical), 20 cm² (non-critical)
  • Total Weight = 11.58 kg
  • Cost = $643.80
  • Fitness = 31.58

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