Free-Form Shape Optimization of Advanced High-Strength Steel Members
Abstract
:1. Introduction
2. Modal analysis and Member Selection
2.1. Elastic Buckling Analysis and Nonlinear Shell FE Models
- -
- Material model: based on the coupon test as shown in Figure 1 of the Q&P1180 AHSS material. The yield strength fy is 1079 MPa, and the Young’s modulus E is 203 GPa.
- -
- Element type: S4R, which is a general-purpose type of shell element that has four nodes (linear formulation) with reduced integration.
- -
- Mesh size: about 10 mm × 15 mm, resulting in a fine mesh. The lip has slightly finer mesh, such as 5 mm × 15 mm. Mesh sensitivity studies were conducted to determine this mesh size, which ensures a reasonable accuracy while balancing computational costs.
- -
- Boundary conditions: globally pinned but warping fixed. Centroid RP-1 and RP-2 are defined as the reference points for end sections. The boundary conditions listed in Figure 2 were applied through the reference points.
- -
- Solution scheme: Rik’s method in ABAQUS; the way of convergence judgment follows the traditional theory [44].
- -
- Geometric imperfections: The initial imperfections are defined according to the one-dimensional spectrum method [45]. For the five initial imperfections in the one-dimensional spectrum method illustrated in Figure 3, based on the existing test results [46], the following amplitudes are adopted: overall buckling imperfection δG,1 = L/2909, δG,2 = L/4010, θG,3 = 0.3 × L/1000; local buckling imperfections δL = 0.75 × t; distortion buckling imperfection δL = 0.31 × t; where L is the member length and t is the cross-section thickness.
- -
- Residual stress: this effect was ignored in the model [47].
2.2. Modal Analysis
2.3. Limiting Factors of Axial Load Capacity
3. Construction of Cross-Section Optimization Model
3.1. Definition of Free-Form Cross-Section in PSO
3.2. Objective Function and Constraint Conditions
- (1)
- The included angle between two adjacent elements shall not be less than 90°. Since the free-form shape of the cross-section in this paper is the complete cross-section obtained by first forming the polyline on the right side of the y-axis, and then mirror folding the y-axis, the first element needs to meet the state shown in Figure 12a, so as to meet the requirement that the included angle between element1 and the folded element is not less than 90°. For other elements, the included angle of adjacent elements shall not be less than 90° according to the restrictions in Figure 12b. The code for this condition in optimization is shown as Equation (12), where θ1 is the corner of the first element, and θi (i = 2, 3, 4, …) is the corner of the subsequent elements:
- (2)
- The member element itself does not cross and tie, as shown in Figure 12c. Note that since the cross-section is a complete cross-section obtained by folding the mirror image of the y-axis, if the one-sided cross-section intersects the y-axis of the symmetry axis, the self-inter cross-section of the complete cross-section will also occur, as shown in Figure 12d, which also needs to be avoided. The code for this condition in optimization is shown as Equations (13) and (14), where xi is the abscissa of the point, (xm, ym) and (xm+1, ym+1) are the end-node coordinates of the m-th element, (xn, yn) and (xn+1, yn+1) are the end-node coordinates of the n-th element, f(x, y) is the functional expression of the m-th element:
3.3. Optimizing Processes and Parameters
4. Optimization Results and FE Verification
4.1. Optimization Path and Optimization Result
4.2. FE Verification
- (1)
- The bearing capacity results of FE simulation are basically consistent with those of PSO optimization.
- (2)
- The PSO-CUFSM in this paper has a remarkable effect. For the free-form shape of cross-sections of FS260, FS300, and FS340, the maximum bearing capacity can be increased by 2.57, 2.05, and 2.43 times, respectively, effectively improving the material utilization of AHSS members.
4.3. Modal Verification
5. Discussion
6. Conclusions
- (1)
- Through nonlinear FE modal identification and classification, it can be seen that the critical stress of the local buckling (L) of the traditional cross-section is low; hence local buckling (L) occurs early, and the interaction with distortional buckling (D) limits the bearing capacity. The optimized cross-section restrains the early occurrence of (L), reduces the coupling effect of local buckling (L) and distortional buckling (D), and thus improves the bearing capacity.
- (2)
- The proposed PSO-DSM method shows promising optimal results, although the DSM was only approximately applied to AHSS sections. Based on the further validation with nonlinear FE results, for the three groups of optimizations, the load bearing capacity was increased by 2.57, 2.05, and 2.43 times, respectively, compared to that of the traditional cross-sections.
- (3)
- The free-form shape optimization mainly restricts the local buckling in the optimization direction. The three groups of optimized cross-sections are similar, and the optimized cross-sections all have web stiffeners, a flange bending inward, and a large area curling outward. In this study, it is defined as “a cross-section similar to Ω shape”.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameter Symbol | Parameter Name | Amplitude | Note |
---|---|---|---|
n_particle | Number of particles | 50 | Number of particles in a single particle swarm (number of cross-sections) |
n_order | Order of the particle | Positive integer | Number of elements of a single particle (number of polyline segments) |
len_element | Length of element | 10 mm | The length of a single polyline segment |
t_element | Thickness of element | 1.2 mm | The actual thickness of the member cross-section, selected from the material properties test of Figure 1 |
w | Weight of inertia | 0.9 | Increase the randomness of motion, selected according to reference [50] |
r1 | Personal learning factors | 1.0 | Increase the randomness of motion, selected according to reference [51] |
r2 | Global learning factors | 2.0 | Increase the randomness of motion, selected according to reference [51] |
max_gen | Maximum number of iterations | 300 | The iteration upper bound, where the optimization terminates. |
pop_range | Motion boundary | (−0.5, 0.5) | Corner of a single round is limited to the interval (−π/2, π/2). |
speed_range | Velocity boundary | (−0.5, 0.5) | Corner of a single round is limited to the interval (−π/2, π/2). |
fy | Yield strength | 1079 MPa | Material parameters of CUFSM, selected from the material properties test of Figure 1 |
E | Elasticity modulus | 203 GPa | Material parameters of CUFSM, selected from the material properties test of Figure 1 |
ID | Iteration k | 1 | 75 | 150 | 225 (Convergence) |
---|---|---|---|---|---|
FS260 | Cross-Section k | ||||
Pk (kN) | 179.16 | 206.55 | 208.61 | 208.92 | |
FS300 | Cross-Section k | ||||
Pk (kN) | 192.37 | 234.46 | 235.08 | 235.10 | |
FS340 | Cross-Section k | ||||
Pk (kN) | 221.05 | 259.06 | 267.26 | 268.42 |
Optimize Grouping | C Cross-Section | Web h (mm) | Flange b (mm) | Hemming a (mm) | Cross-Section Sketch |
---|---|---|---|---|---|
FS260 | TC260-1 | 160 | 35 | 15 | |
TC260-2 | 120 | 50 | 20 | ||
FS300 | TC300-1 | 180 | 45 | 15 | |
TC300-2 | 140 | 60 | 20 | ||
FS340 | TC340-1 | 200 | 55 | 15 | |
TC340-2 | 160 | 70 | 20 |
Specimen | Pmax, PSO (MPa) | Pmax, FEM (MPa) | Variation (Pmax, PSO–Pmax, FEM)/Pmax, PSO |
---|---|---|---|
FS260 | 208.92 | 205.65 | 1.57% |
FS300 | 235.10 | 210.50 | 10.46% |
FS340 | 268.42 | 285.87 | −6.50% |
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Yin, L.; Deng, T.; Niu, Y.; Li, Z. Free-Form Shape Optimization of Advanced High-Strength Steel Members. Buildings 2022, 12, 2101. https://doi.org/10.3390/buildings12122101
Yin L, Deng T, Niu Y, Li Z. Free-Form Shape Optimization of Advanced High-Strength Steel Members. Buildings. 2022; 12(12):2101. https://doi.org/10.3390/buildings12122101
Chicago/Turabian StyleYin, Lingfeng, Tianyang Deng, Yu Niu, and Zhanjie Li. 2022. "Free-Form Shape Optimization of Advanced High-Strength Steel Members" Buildings 12, no. 12: 2101. https://doi.org/10.3390/buildings12122101