ENGR 5005G: Advanced Topology Optimization

Assessment 2

Questions

1. Consider the structure in the following Figure. Determine the displacements of node 1. All three bars are made by same material with the same circular cross section Area. Record the displacement values in x, and y directions, stress in the bars, and the total weight of the structure.

Change the cross-sectional areas to minimize the total weight. Each bar may have a different radius now. Set the maximum and minimum acceptable radius of bars as the manufacturing constraints to 30 mm and 8 mm, respectively. Try to find the minimum weight of the structure while maintain: 

• the displacements in x-direction within 5?viation
• the displacements in Y-direction within 5?viation 
• The total magnitude of displacement vector in x-y plane within 5?viation

2. Consider the angle defined as 45 degrees, as your fourth variable that can be anything between 30 to 60 degrees. Set the maximum and minimum acceptable values as the manufacturing constraints. Find the best design for the following three cases maintaining:

• the displacements in x within 5?viation 
• the displacements in Y within 5?viation 
• The total displacement vector within 5?viation 

Assessment Requirements – Summary

Objective:
The assessment focuses on structural optimization of a truss-like system to determine node displacements, stresses, and total weight while considering material and geometric constraints. The student is required to:

Key Tasks:

1. Node Displacement Analysis:

   • Calculate displacements at Node 1 in both x and y directions.
   • Determine stress in each bar.
   • Record the total weight of the structure.

2. Structural Optimization:

   • Modify the cross-sectional areas of the bars to minimize total weight.
   • Apply manufacturing constraints: minimum radius = 8 mm, maximum radius = 30 mm.
   • Ensure that displacements remain within ±5?viation in x, y, and total displacement magnitude.

3. Angle Optimization:

   • Consider the fourth variable: angle = 45° initially, adjustable within 30°–60°.
   • Determine the optimal combination of cross-sectional areas and angle for three different design cases.
   • Maintain the displacement limits within ±5?viation.

Deliverables:

• Tabulated displacements in x and y directions for Node 1.
• Stress values for all bars.
• Total weight of the structure before and after optimization.
• Optimized design variables (bar radii and angle).

Assessment Approach – Mentor Guidance

The Academic mentor guided the student step by step through the following process:

1. Understanding the Problem:

   • Explained the truss structure, nodes, bars, and the concept of displacement and stress analysis.
   • Emphasized the importance of material uniformity and geometric constraints.

2. Initial Analysis:

   • Step 1: Calculate initial displacements at Node 1 using classical truss analysis (e.g., finite element or matrix method).
   • Step 2: Compute the stress in each bar based on the given circular cross-section and material properties.
   • Step 3: Record the total weight of the structure using the formula:W=∑(density×volume of each bar)W = \sum (\text{density} \times \text{volume of each bar})W=∑(density×volume of each bar)

3. Optimization of Bar Radii:

   • Explained how to adjust cross-sectional areas while maintaining displacement constraints.
   • Introduced the concept of iterative optimization, adjusting each bar’s radius within 8–30 mm.
   • Verified that x and y displacements, and total displacement vector, stayed within ±5?viation.

4. Angle Optimization:

   • Mentored the student to consider the angle as the fourth design variable within the range 30°–60°.
   • Used either analytical or software-assisted methods to evaluate the combined effect of angle and bar radii on displacement and weight.
   • Checked all three design cases for displacement limits compliance.

5. Validation and Final Output:

   • Compiled results: final displacements, stresses, and optimized weights.
   • Ensured all manufacturing constraints and displacement tolerances were satisfied.
   • Tabulated and documented all final design variables for clarity.

Outcome & Learning Objectives Covered

Outcome Achieved:

• Successfully calculated displacements, stresses, and weight for the initial structure.
• Identified the optimal combination of bar radii and angle to minimize weight while meeting all displacement constraints.
• Produced a clear, tabulated summary of the optimized design for decision-making or further analysis.

Learning Objectives Covered:

1. Apply fundamental structural analysis techniques to calculate displacements and stresses.
2. Understand and implement design optimization within practical constraints.
3. Analyze the impact of geometric and material variations on structural performance.
4. Integrate iterative problem-solving and software tools for engineering design decisions.
5. Develop skills in reporting and documenting structural optimization results clearly.

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