Parametric FEA Analysis: Stress Concentration in Perforated Plates

Automated finite element analysis workflow for studying stress concentrations in plates with circular holes under tensile loading

Overview

This project presents a comprehensive parametric study investigating stress concentration factors in aluminum plates with circular holes subjected to uniaxial tension. An automated Python-based workflow was developed to systematically analyze 125 geometric configurations, validating FEA results against Peterson's theoretical predictions with excellent accuracy (average error < 5%).

Key Features

• Automated parametric FEA pipeline using FreeCAD and Python
• Mesh convergence study demonstrating solution stability (<2% variation)
• Theoretical validation against Peterson's stress concentration factor approximation
• Comprehensive visualization suite including heatmaps, Pareto plots
• Design optimization insights for weight-stress trade-offs

Objectives

1. Validate FEA methodology against established theoretical predictions
2. Quantify the effects of geometric parameters (hole diameter, plate width, thickness) on stress concentration
3. Develop a reusable computational workflow for parametric mechanical analysis
4. Provide engineering design guidelines for perforated plate applications

Sample Visualizations

Methodology

Problem Setup

Geometry:

• Rectangular aluminum plate (L × W × t)
• Center circular hole (diameter d)
• Fixed constraint on one end
• Tensile load P = 50 kN on opposite end

Material: Aluminum 6061-T6

• Young's Modulus: 69,000 MPa
• Poisson's Ratio: 0.33
• Yield Strength: 276 MPa

Parameter Ranges:

• Hole diameter (d): 10 - 50 mm (5 levels)
• Plate width (W): 60 - 200 mm (5 levels)
• Thickness (t): 5 - 15 mm (5 levels)

Theoretical Background

Nominal stress in net section:

σ_nom = P / ((W - d) × t)

Peterson's approximation for K_t:

K_t = 3 - 3.13(d/W) + 3.66(d/W)² - 1.53(d/W)³

Maximum stress:

σ_max = K_t × σ_nom

Mesh Convergence Study

Four mesh refinement cases analyzed, demonstrating convergence:

Case	Element Size	Max σvM (MPa)	Variation
1	5.00 mm (Moderate)	213	-0.9%
2	2.00 mm (Fine)	217	+0.9%
3	5.00 mm (Fine on hole)	213	-0.9%
4	3.00 mm (Fine)	216	+0.5%

Average FEA result: 214.75 MPa
Theoretical prediction: 215 MPa
Selected mesh: 3.00 mm for optimal balance of accuracy and computational cost

Repository Structure

parametric-fea/
├── README.md                            
├── requirements.txt                    
├── src/
│   ├── generate_results.py             
│   └── analyze_params.py               
│
├── models/
│   ├── plate_n.FCStd                   
│  
├── data/
│   └── results_plate_hole_c.csv        
│
└── plots/
    ├── kt_fea_vs_theory.png
    ├── stress_vs_holediameter.png
    ├── stress_vs_thickness.png
    ├── heatmap_stress.png
    ├── pareto_stress_vs_weight.png

Requirements

• FreeCAD 0.20+: Download here
• Python 3.7+: Download here
• FreecadParametricFEA wrapper: GitHub repo

Installation

1. Clone the repository:

git clone https://github.com/DennisxB/parametric-fea.git
cd parametric-fea-study

2. Install Python dependencies:

pip install -r requirements.txt

3. Configure FreeCAD path:

Edit src/generate_results.py and update:

FREECAD_PATH = "C:/FreeCAD-0.20/bin"  # Adjust to your installation

Usage

1. Run Parametric FEA Analysis

python src/generate_results.py

2. Analyze Results and Generate Plots

python src/analyze_params.py

Model Validation

• FEA vs Theory correlation: R² > 0.95
• Average K_t error: 3.94%

Parametric Sensitivity

Stress increases sharply with hole diameter — doubling d can raise σ_max by 3–5 times, with a critical point near d/W ≈ 0.5. Increasing plate width W reduces stress, and it’s more effective than increasing thickness, though gains taper off beyond W > 150 mm. Thickness t shows a simple, linear effect and doubling t roughly halves the peak stress, making it the most predictable parameter.

Validation Against Theory from Sample Results

d/W Ratio	Kt (Theory)	Kt (FEA)	Error (%)
0.10	2.60	2.63	1.2
0.25	2.44	2.47	1.2
0.50	2.15	2.13	-0.9
0.75	2.03	2.05	1.0

Contributions

Contributions are welcome! Areas for improvement:

• Add support for non-circular holes (elliptical, rectangular)
• Implement multi-hole configurations
• Add fatigue life prediction
• Develop optimization algorithms (genetic algorithm, gradient-based)
• Create interactive Plotly dashboards
• Add experimental validation data
• Extend to composite materials

References

1. R. E. Peterson, Stress Concentration Factors. New York: Wiley, 1974.

2. W. D. Pilkey and D. F. Pilkey, Peterson's Stress Concentration Factors, 3rd ed. Hoboken, NJ: Wiley, 2008.

3. O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, 6th ed. Oxford: Butterworth-Heinemann, 2005.

4. W. C. Young and R. G. Budynas, Roark's Formulas for Stress and Strain, 7th ed. New York: McGraw-Hill, 2002.

5. R. G. Budynas and J. K. Nisbett, Shigley's Mechanical Engineering Design, 9th ed. New York: McGraw-Hill, 2011.

6. FreeCAD Community, "FreeCAD: Your own 3D parametric modeler," 2021. [Online]. Available: https://www.freecadweb.org/