Scaling the Dynamic Buckling Behavior of a Box Girder Based on the Finite Similitude Approach
Abstract
1. Introduction
2. Finite Similitude Theory
2.1. Transport Equations
2.2. Derivation of Similar Equations
2.3. Scaling of the Constitutive Equations
2.4. Practical Constraints
3. Similarity Criteria for the Box Girder Based on Finite Similitude Theory
3.1. Scaling Criteria for Static Ultimate Strength of Box Girders
- (1)
- Determine the geometric dimensions lps, material properties (density ρps, material yield stress σyps), and boundary conditions of the full-scale model of the box girder;
- (2)
- Determine the geometric scale factor β, material properties (density ρts, material yield stress σyts), and boundary conditions of the box girder trial model;
- (3)
- The geometric dimensions lts and density scale factor αρ of the trial model are determined by similarity criteria Π1 and Π2, respectively;
- (4)
- The time scale factor h is calculated by the similarity criterion Π3;
- (5)
- The bending moment M and the corresponding rotation angle θ of the full-scale model and the trial model of the box girder are obtained through experimental or finite element calculation, and the predicted bending moment Mps and angle θps relative to the full-scale model are calculated using the similarity criteria Π4 and Π5.
3.2. Scaling Criteria for Dynamic Responses of Box Girders
- (1)
- Determine the geometric dimensions lps, material properties (density ρps, material yield stress σyps), initial conditions (initial peak bending moment Mdps, bending moment action time history Tdps), and boundary conditions of the full-scale model of the box girder;
- (2)
- Determine the geometric scale factor (β), material properties (density ρts, material yield stress σyts), initial conditions (initial peak load Mdts, load action time history Tdts), and boundary conditions of the box girder trial model;
- (3)
- The geometric dimensions (lts) and density scale factor (αρ) of the trial model are determined using similarity criteria Π1 and Π2, respectively;
- (4)
- The time scale factor h is calculated by the similarity criterion Π3;
- (5)
- Obtain the displacement angle (θ) and its temporal evolution (time history, t) for both the full-scale and trial models of the box girder through experimental or finite element calculations, and calculate the predicted displacement angle (θps) and the change in the angle over time relative to the full-scale model using similarity criteria Π4 and Π5.
3.3. Correction of the Nonlinear Response Scaling Criterion of Box Girder
- (1)
- Determine the geometric dimensions lps, material properties (density ρps, constitutive relationship, such as the C-S equation), initial conditions (initial peak bending moment load Mdps, initial load action time history Tdps), and boundary conditions of the full-scale model of the box girder;
- (2)
- Determine the geometric scale factor β, the material properties (density ρts, constitutive relationship C-S equation), and boundary conditions of the box girder trial model;
- (3)
- The geometric dimensions lts and density scale factor αρ of the trial model are determined by similarity criteria Π1 and Π2, respectively;
- (4)
- Determine the average dynamic stress between the full-scale model of the box girder and the materials of each trial model, calculate the velocity ratio βv using the similarity criterion Π3, and further obtain the time scale factor h through the similarity criterion Π4;
- (5)
- Calculate the time history Tts and initial peak bending moment load Mdts of the box girder trial model by combining the scale factors Π5 and Π6, respectively;
- (6)
- The displacement angle θ of the full-scale model of the box girder and the trial model are obtained by experiment or finite element calculation, and the predicted displacement angle θps and displacement-time change history relative to the full-scale model are calculated using the similarity criteria Π5 and Π7.
4. Numerical Analyses
4.1. Finite Element (FE) Model of Box Girder Specimen
4.2. Boundary and Loading Conditions
4.2.1. Static Load Setting
4.2.2. Dynamic Load Setting
4.3. Design Results of Small-Scale Box Girder Model
4.3.1. Detailed Information
4.3.2. Modified Scaling Factors
4.4. Mesh Convergence Analysis
5. Similarity Validation of Quasi-Static Cases
5.1. Ultimate Strength
5.2. Buckling Failure Mode
6. Similarity Validation of Dynamic Cases
6.1. Dynamic Critical Buckling Load
6.2. Correction of Dynamic Critical Buckling Load
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Symbol | Meanings |
---|---|
D* | Differential |
t | Time |
Ω | Control Volume |
ps | Physical Space |
ts | Trial Space |
ρ | Density |
Ψ | Physical Quantity |
V | Unit Volume |
v | Velocity |
n | Unit Normal Of The Boundary Γ Of The Control Volume Ω |
Γ | The Boundary For Ω |
J | Flux |
b | Source Term |
αρ | Density Scaling Factor |
β | Scaling Ratio |
h | Time Scaling Factor |
σy/σstat | Static Yield Strength |
αv | Velocity Scaling Factor |
αu | Displacement Scaling Factor |
u | Displacement |
l | Scale |
σi | Unit Stress |
Fi | Unit Force |
Ai | Unit Area |
F | Force |
M | Bending Moment |
L | Arm Of Force |
θ | Angle Of Rotation |
R | Radius Of Corner |
Average Stress | |
ε | Strain |
D | Coefficient In C—S Constitutive Equation |
q | Coefficient In C—S Constitutive Equation |
βv | Velocity Modifying Factor |
ζ | Ratio Of Average Dynamic Yield Strength |
M0 | Initial Bending Moment Load |
T0 | Initial Loading Time |
hmod | Modified Time Scaling Factor |
M0mod | Modified Bending Moment Load |
T0mod | Modified Loading Time |
NO (Material) | Geometry Properties | Material Properties | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Scale Ratio | Plate | Stiffener | Plate and Stiffener | ||||||||||
β | a | b | t | hs | ts | hw | tw | ρ (10−9) (T/mm3) | μ | E (GPa) | σy (MPa) | σy/E (10−3) | |
Full-scale model (Q235 [36]) | 1 | 600 | 400 | 5 | 60 | 4 | 120 | 8 | 7.85 | 0.3 | 206 | 235 | 1.14 |
Trial model 1 (Q235) | 2 | 300 | 200 | 2.5 | 30 | 2 | 60 | 4 | 7.85 | 0.3 | 206 | 235 | 1.14 |
Trial model 2 (Q345 [37]) | 2 | 300 | 200 | 2.5 | 30 | 2 | 60 | 4 | 7.85 | 0.3 | 206 | 345 | 1.68 |
Trial model 3 (Al [38]) | 2 | 300 | 200 | 2.5 | 30 | 2 | 60 | 4 | 2.69 | 0.33 | 72 | 295 | 4.07 |
Trial model 4 (S235 [39]) | 2 | 300 | 200 | 2.5 | 30 | 2 | 60 | 4 | 7.80 | 0.3 | 210 | 240 | 1.14 |
Trial model 5 (20 steel [36]) | 2 | 300 | 200 | 2.5 | 30 | 2 | 60 | 4 | 7.85 | 0.29 | 211 | 245 | 1.16 |
Location | Translation | Rotational | ||||
---|---|---|---|---|---|---|
ux | uy | uz | θx | θy | θz | |
RP1 | fix | fix | - | fix | 0.1/Md | fix |
RP2 | fix | fix | fix | fix | −0.1/Md | fix |
M0 | T0 | |
---|---|---|
Full-scale model | 1.87 × 109 | 7.71 × 10−3 |
Trial model 1 | 2.34 × 108 | 3.85 × 10−3 |
Trial model 4 | 2.39 × 108 | 3.80 × 10−3 |
Trial model 5 | 2.41 × 108 | 3.82 × 10−3 |
Q235-Q235 | 2 | 0.125 | 2.000 | 0.125 | 0.063 |
Q235-Q345 | 2 | 0.125 | 2.423 | 0.151 | 0.063 |
Q235-Al | 2 | 0.043 | 3.832 | 0.082 | 0.021 |
Q235-S235 | 2 | 0.124 | 2.028 | 0.126 | 0.062 |
Q235-20 steel | 2 | 0.125 | 2.042 | 0.128 | 0.063 |
D (s−1) | q | |
---|---|---|
Full-scale model | 40.4 | 5 |
Trial model 1 | 40.4 | 5 |
Trial model 2 | 40.4 | 5 |
Trial model 3 | 12,880 | 4 |
Trial model 4 | 4000 | 5 |
Trial model 5 | 40.4 | 5 |
Q235-Q235 | 2 | 1.03 | 0.125 | 2.06 |
Q235-S235 | 2 | 0.90 | 0.124 | 1.80 |
Q235-20 steel | 2 | 1.05 | 0.125 | 2.10 |
M0 | M0mod | T0 | T0mod | |
---|---|---|---|---|
Full-scale model | 1.87 × 109 | 1.87 × 109 | 7.71 × 10−3 | 7.71 × 10−3 |
Trial model 1 | 2.34 × 108 | 2.48 × 108 | 3.85 × 10−3 | 3.74 × 10−3 |
Trial model 4 | 2.39 × 108 | 1.88 × 108 | 3.80 × 10−3 | 3.28 × 10−3 |
Trial model 5 | 2.41 × 108 | 2.58 × 108 | 3.82 × 10−3 | 3.67 × 10−3 |
Mtsmax | Mpsmax | Error | θts | θps | Error | |
---|---|---|---|---|---|---|
Full-scale model | —— | 1.87 × 109 | —— | —— | 9.50 × 10−4 | —— |
Trial model 1 | 2.34 × 108 | 1.87 × 109 | 0.00% | 9.50 × 10−4 | 9.50 × 10−4 | 0.00% |
Trial model 2 | 3.05 × 108 | 1.66 × 109 | −11.11% | 1.25 × 10−3 | 1.25 × 10−3 | −31.55% |
Trial model 3 | 2.18 × 108 | 1.39 × 109 | −25.57% | 2.88 × 10−3 | 2.88 × 10−3 | 203.30% |
Trial model 4 | 2.39 × 108 | 1.87 × 109 | 0.02% | 9.50 × 10−4 | 9.50 × 10−4 | 0.05% |
Trial model 5 | 2.41 × 108 | 1.85 × 109 | −1.07% | 9.63 × 10−4 | 9.63 × 10−4 | 1.42% |
Dynamic Buckling Critical Load | ||||||||
---|---|---|---|---|---|---|---|---|
Td = T0 | Error | Td = 1.33T0 | Error | Td = 2T0 | Error | Td = 4T0 | Error | |
Full-scale model | 1.90 | \ | 1.70 | \ | 1.51 | \ | 1.30 | \ |
Trial model 1 | 1.95 | −2.63% | 1.74 | −2.35% | 1.55 | −2.65% | 1.34 | −3.08% |
Trial model 4 | 1.65 | 13.16% | 1.49 | 12.35% | 1.34 | 11.26% | 1.19 | 8.46% |
Trial model 5 | 1.95 | −2.63% | 1.75 | −2.94% | 1.56 | −3.31% | 1.35 | −3.85% |
Dynamic Buckling Critical Load | ||||||||
---|---|---|---|---|---|---|---|---|
Td = T0 | Error | Td = 1.33T0 | Error | Td = 2T0 | Error | Td = 4T0 | Error | |
Full-scale model | 1.90 | \ | 1.70 | \ | 1.51 | \ | 1.30 | \ |
trial model 1 | 1.92 | −1.05% | 1.71 | −0.59% | 1.53 | −1.32% | 1.31 | −0.77% |
trial model 4 | 1.77 | 6.84% | 1.59 | 6.47% | 1.43 | 5.30% | 1.27 | 2.31% |
trial model 5 | 1.91 | −0.53% | 1.71 | −0.59% | 1.52 | −0.66% | 1.31 | −0.77% |
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Xu, C.; Wang, Z.; Kong, X.; Zhou, H.; Zheng, C.; Wu, W. Scaling the Dynamic Buckling Behavior of a Box Girder Based on the Finite Similitude Approach. J. Mar. Sci. Eng. 2025, 13, 1496. https://doi.org/10.3390/jmse13081496
Xu C, Wang Z, Kong X, Zhou H, Zheng C, Wu W. Scaling the Dynamic Buckling Behavior of a Box Girder Based on the Finite Similitude Approach. Journal of Marine Science and Engineering. 2025; 13(8):1496. https://doi.org/10.3390/jmse13081496
Chicago/Turabian StyleXu, Chongxi, Zhuo Wang, Xiangshao Kong, Hu Zhou, Cheng Zheng, and Weiguo Wu. 2025. "Scaling the Dynamic Buckling Behavior of a Box Girder Based on the Finite Similitude Approach" Journal of Marine Science and Engineering 13, no. 8: 1496. https://doi.org/10.3390/jmse13081496
APA StyleXu, C., Wang, Z., Kong, X., Zhou, H., Zheng, C., & Wu, W. (2025). Scaling the Dynamic Buckling Behavior of a Box Girder Based on the Finite Similitude Approach. Journal of Marine Science and Engineering, 13(8), 1496. https://doi.org/10.3390/jmse13081496