VM-DX-MECH-003
VM-DX-MECH-003
Optimization of Water Tank Column for Mass and Natural
Frequency
Overview
| Reference: | S. S. Rao, Optimization Theory and Application Second edition, example 1.10, page 28-30 |
| Analysis Type(s): | Goal Driven Optimization with APDL |
| Element Type(s): | 3-D Solid |
Test Case
A uniform column of rectangular cross section b and d m is to be constructed for supporting a water tank of mass M. It is required to:
minimize the mass of the column for economy
maximize the natural frequency of transverse vibration of the system for avoiding possible resonance due to wind.
Design the column to avoid failure due to direct compression (should be less than maximum permissible compressive stress) and buckling (should be greater than direct compressive stress). Assume the maximum permissible compressive stress as σmax. The design vector is defined as:
where:
| b = width of cross-section of column |
| d = depth of cross-section of column |
Input Parameters: — Width and Height
Response Parameters: — Mass, Natural Frequency, Direct Stress, Buckling Stress
| Material Properties | Geometric Properties | Loading |
|
E = 3e10 Pa ρ = 2300 Kg/m3 σ max = 4.1e7 Pa |
Width, b = 0.4 m Depth, d =1.2 m Length, I = 20 m |
Mass of water tank M = 1000000 Kg Acceleration due to gravity = 9.81 m/s2 |
Sample Size: — 10000
| Results | Target | DesignXplorer | Error (%) |
|---|---|---|---|
| Width b | 0.36102 m | 0.36102 m | 0.000 |
| Depth d | 1.3181 m | 1.318137 m | 0.002 |
| Mass of column M | 21890 kg | 21889.77 kg | –0.001 |
| Natural frequency w | 0.87834 rad/sec | 0.87816 rad/sec | -0.020 |
| Direct stress | 2.0386e7 Pa | 2.0383e7 Pa | -0.015 |
| Buckling stress | 6.1526e6 Pa | 6.15174e6 Pa | -0.013 |
Analysis
Minimize:
Maximize:
Subject to constraints:
Required objective is obtained by having:
| b = 0.36102 m |
| d = 1.3181 m |
| M = (minimum) = 21890 kg |
| W = (maximum) = 0.87834 rad/sec |
| Direct stress = 2.0386e7 Pa |
| Buckling stress = 6.1526e6 Pa |
Results Comparison
| Results | Target | DesignXplorer | Error (%) |
|---|---|---|---|
| Width b | 0.36102 m | 0.36102 m | 0.000 |
| Depth d | 1.318100 m | 1.318137 m | 0.003 |
| Mass of column M | 21890.0000 kg | 21890.20 kg | 0.001 |
| Natural frequency w | 0.87834 rad/sec | 0.87815 rad/sec | -0.022 |
| Direct stress | 20386000 Pa | 2.0385e7 Pa | -0.006 |
| Buckling stress | 6152600 Pa | 6.15260e6 Pa | -0.023 |