A Modified Catenary Model with Application to the Analysis and Design of Retrofit Cables for Progressive Collapse
Abstract
:1. Introduction
2. Modified Catenary Behavior
2.1. Load-Deflection Equations
2.2. Approximate Static Load-Deflection Equations for Design
2.3. Dynamic Load Deflection Equations Using Dynamic Amplification Factors
3. Materials, Methods, and Experimental Setup
3.1. Finite Element Model of M.C. System
3.2. Static Experiments
3.3. Dynamic Experiments
4. Results and Discussion
4.1. Analytical Results
4.2. Static Experiment Results
4.3. Dynamic Experiment Results
5. Application to Retrofit Cable Design
5.1. Retrofit Design Equations and Procedures
- Establish a given deflection limit, or (e.g., , floor-to-floor height, ultimate strain)
- Calculate from a given deflection limit and from Equation (17)
- If < 1, use Equation (21), and if > 1, use Equation (22).
5.2. Illustrative Cable Retrofit Behavior Deisgn Example
- Case 1
- Case 2
5.3. Limitations of the Method
6. Summary and Conclusions
- As expressed in Equation (11), geometric nonlinearities in modified catenary systems in the elastic material range result in load carrying capacity that is approximately proportional to the cubic displacement.
- As expressed in Equation (12), geometric nonlinearities in modified catenary systems provide significant post-yield load carrying capacity that is approximately proportional to post-yield displacement.
- For a given displacement, the ratio of dynamic load to static load is ¼ for the elastic case and varies with post yield.
- Static and dynamic experimental results verify the applicability of modified catenary behavior for scaled cable systems (i.e., 16-gage steel wire with 680 mm spans).
- The approximate modified catenary equations adequately represent the essential features of a retrofit cable system and provide closed-form estimations of required cross-sectional areas to be used in design.
Author Contributions
Funding
Conflicts of Interest
References
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s (mm) | A (mm2) | E (GPa) | Fy (MPa) | |
---|---|---|---|---|
Configuration 1 (Small Scale) | 340 | 1.5 | 200 | 460 |
Configuration 2 (Large Scale) | 6.1 × 103 | 3.2 × 103 | 97 | 830 |
Test | δ0 (mm) | fn (Hz) | Tn (s) | a+ (g) | a− (g) | u+ (mm) | u− (mm) | ||||
---|---|---|---|---|---|---|---|---|---|---|---|
M.C. | Exp. | M.C. | Exp. | M.C. | Exp. | M.C. | Exp. | M.C. | M.C. | ||
Elastic 1 | −1.50 | 7.99 | 8.90 | 0.13 | 0.11 | 2.62 | 3.05 | −1.00 | −1.03 | −1.50 | −15.88 |
Elastic 2 | −0.29 | 7.99 | 8.40 | 0.13 | 0.12 | 2.81 | 1.88 | −1.01 | −1.10 | −0.29 | −16.11 |
Elastic 3 | −0.50 | 7.99 | 7.95 | 0.13 | 0.13 | 2.76 | 1.60 | −1.00 | −0.96 | −0.50 | −16.04 |
Elastic 4 | −1.81 | 7.99 | 9.05 | 0.13 | 0.11 | 2.49 | 2.66 | −1.00 | −1.10 | −1.81 | −15.71 |
Elastic 5 | −1.61 | 7.99 | 7.60 | 0.13 | 0.13 | 2.52 | 1.70 | −1.00 | −0.89 | −1.61 | −15.75 |
Inelastic 1 | −3.80 | 6.99 | 7.45 | 0.14 | 0.13 | 1.90 | 1.77 | −1.03 | −1.56 | −3.80 | −23.03 |
Inelastic 2 | −0.41 | 7.99 | 8.45 | 0.13 | 0.12 | 2.05 | 1.50 | −1.13 | −1.10 | −0.41 | −24.24 |
Inelastic 3 | −0.25 | 7.99 | 8.65 | 0.13 | 0.12 | 2.06 | 1.37 | −1.13 | −1.20 | −0.25 | −24.32 |
Inelastic 4 | −6.60 | 6.99 | 7.35 | 0.14 | 0.14 | 1.84 | 1.64 | −1.00 | −1.31 | −6.60 | −22.54 |
Inelastic 5 | −1.61 | 6.99 | 6.95 | 0.14 | 0.14 | 1.99 | 1.82 | −1.08 | −0.98 | −1.61 | −23.72 |
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Deputy, L.T.; Zeinali, Y.; Story, B.A. A Modified Catenary Model with Application to the Analysis and Design of Retrofit Cables for Progressive Collapse. Infrastructures 2018, 3, 26. https://doi.org/10.3390/infrastructures3030026
Deputy LT, Zeinali Y, Story BA. A Modified Catenary Model with Application to the Analysis and Design of Retrofit Cables for Progressive Collapse. Infrastructures. 2018; 3(3):26. https://doi.org/10.3390/infrastructures3030026
Chicago/Turabian StyleDeputy, Leven T., Yasha Zeinali, and Brett A. Story. 2018. "A Modified Catenary Model with Application to the Analysis and Design of Retrofit Cables for Progressive Collapse" Infrastructures 3, no. 3: 26. https://doi.org/10.3390/infrastructures3030026