Figure 1:
Solution methodology flow chart
Figure 2:
Frame system units
Figure 3:
Fault tree diagram for the frame system unit
Figure 4:
Reliability block diagram for the frame system unit
Table 1:
Structural components notations
| Component Notation | Component Failure | Component Notation | Component Failure | ||
|---|---|---|---|---|---|
| x1 | Flexural failure | Beam failure | x5 | Floor system failure | |
| x2 | Shear failure | x6 | Base Plate failure | ||
| x3 | Bolts failure | x7 | Stability failure | Column failure | |
| x4 | Weld failure | x8 | Material failure | ||
Table 2:
The importance of component (1) for a three-system component (Rausand & Hoyland, 2003)
| (.,x2,x3) | ϕ(1, x2, x3) − ϕ(0, x2, x3) | C(11,x2,x3) |
|---|---|---|
| (.00) | 0 | |
| (.01) | 1 | {1,3} |
| (.10) | 1 | {1,2} |
| (.11) | 1 | {1,2,3} |
Table 3:
The importance of all the components for the frame system
| Component | Component importance Bϕ(xn) | Component | Component importance Bϕ(xn) |
|---|---|---|---|
| x1 | 0.0234375 | x5 | 0.3515625 |
| x2 | 0.0234375 | x6 | 0.3515625 |
| x3 | 0.0234375 | x7 | 0.1171875 |
| x4 | 0.0234375 | x8 | 0.1171875 |
Figure 5:
Constraints violation and final objective function for frame system elements
Table 4:
Objective function and constraints violation of the whole unified system
| Iterations | Objective Function | Constraints Violation |
|---|---|---|
| 0 | 3114.32 | 0.3029 |
| 1 | 2983.31 | 0 |
| 2 | 2438.8 | 0 |
| 3 | 2435.23 | 0 |
Table 5:
Importance factor effect on optimally designed frame system components
| Frame components | Independent Failure of separated components | Independent Failure of the whole system | % Difference | |
|---|---|---|---|---|
| 1 | Number of the studs | 43 | 45 | −4.65 |
| 2 | Diameter of the stud [in] | 0.5604 | 0.5 | 10.778 |
| 3 | Plastic section modulus of the beam [in3] | 245.7 | 245.7 | 0.0 |
| 4 | Cross-sectional area of the beam [in2] | 4.1 | 4.3 | −4.878 |
| 5 | Moment of inertia of the beam [in4] | 1885.3 | 1885.4 | −0.0053 |
| 6 | Moment of inertia of the column [in4] | 89.2383 | 89.2 | 0.0429 |
| 7 | Cross-sectional area of the column [in2] | 23 | 22.9 | 0.435 |
| 8 | Base plate width [in] | 14.7169 | 14.4 | 2.153 |
| 9 | Base plate length [in] | 14.7582 | 15.4 | 4.348 |
| 10 | Base plate thickness [in] | 1.3627 | 1.3 | 4.6 |
Table 6:
Survivor function with and without the importance factor
| Time Unit | Survivor function without the importance factor effect | Survivor function with importance factor effect | % Reduction due to the importance factor impact |
|---|---|---|---|
| 0 | 1 | 1 | NA |
| 0.5 | 0.6959 | 0.6266 | 10 % lower |
| 1 | 0.40375 | 0.3133 | 22.5 % lower |
| 1.5 | 0.2074 | 0.13186 | 36.5 % lower |
| 2 | 0.0927 | 0.0494 | 47 % lower |
Figure 6:
Survivor function of frame system unit with time including and excluding the structural importance factor effect