Analytical Pushover Curves for X-Concentric Braced Steel Frames
Abstract
:1. Introduction
2. Trilinear Proposed Curves
- In phase 1, the X-CBF has a high lateral stiffness ( ), everything remains elastic, and there are no buckling phenomena;
- In phase 2, the structure loses about half of its lateral stiffness because of the buckling of the compressed diagonal; by neglecting the contribution of this one, the lateral stiffness becomes ;
- In phase 3, the system loses all its stiffness () because of the yielding of the diagonal in tension.
- : shear base force;
- : top displacement;
- : Young’s modulus;
- : diagonal cross-section;
- : beam–column angle;
- : diagonal length.
- : reduction coefficient for stability problems;
- : normalized slenderness of the diagonal;
- : imperfection coefficient;
- : reduction coefficient for local buckling.
- (1)
- Instantaneous loss of load;
- (2)
- Relevant softening after the elastic phase;
- (3)
- Plastic behavior caused by relevant post-critical resources of the compressed diagonal.
- A squat diagonal () will show a behavior similar to that of curve (3);
- A slender diagonal () will show a behavior like curve (1), typical of cables, for example;
- A medium-slender diagonal () will show an intermediate behavior, like curve (2).
- If the load discharge is instantaneous (line (1) in Figure 4), a vertical line is followed, until the intersection with the line with stiffness in correspondence to the shear value ;
- If the compressed diagonal has a pseudo-plastic behavior, line (3) is followed, with a slope equal to =/2, reaching the third phase in correspondence to shear value and displacement;
- If the compressed diagonal has medium slenderness (), it is possible to reach the third phase with curve (2) characterized by a plastic shear lower than .
- The “Lower-bound” curve, characterized by a second phase with a stiffness less than half of , and achievable by connecting the point in correspondence to with the one with
- The “Upper-bound” curve, characterized by a higher second-phase stiffness, equal to , and with a plastic shear value equal to .
3. Numerical Validation
4. Numerical Comparisons
5. Application to Mono- and Multi-Storey Buildings
6. Case Studies
- Mono- or multi-storey buildings;
- Behavior factor equal to 1 or 4.
6.1. One-Floor Case Studies
6.2. Multi-Story Case Studies
7. Conclusions
- It was first observed that the lateral stiffness of the X-CBFs in the Post-Buckling phase is not equal to half of the Pre-Buckling phase stiffness but is smaller and variable with the normalized slenderness of the diagonal.
- Taking these aspects into account, the construction of an analytical pushover “spindle” was proposed, characterized by lower- and upper-bound curves. Such limit curves form a field in which numerical pushover curves have to fall. This was proven by using a model validated through experimental tests. By varying the diagonal profiles with different normalized slenderness, it was shown that numerical curves fall always inside the proposed pushover spindle. In addition, the higher the normalized slenderness of the diagonal is, the tighter the analytical spindle is.
- The proposal was then extended to whole structures by analyzing mono- and multi-storey buildings. The numerical response of all the analyzed case studies falls back inside the spindle, which represents a strict strength domain and allows correctly evaluating the behavior of these types of structures.
- The proposed approach can be directly used in the design phase within the pushover method and allows a possible analytical control on the results obtained with the numerical model.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Drift | Performance Level | ||
---|---|---|---|
S5 | S3 | S1 | |
Transitional | 2% | 1.5% | 0.7% |
Permanent | 2% | 0.5% | negligible |
Geometrical Characteristics | Mechanical Characteristics | |||||||
---|---|---|---|---|---|---|---|---|
Element | (mm) | (mm) | (mm) | (mm) | (mm) | (MPa) | ||
Beam | 250 | 125 | 6 | 9 | 286 | 204,286 | 0.012 | 0.247 |
Column | 175 | 175 | 7.5 | 11 | 266 | 204,615 | 0.0015 | 0.306 |
Diagonal | 100 | 50 | 4 | 6 | 325 | 216,667 | 0.011 | 0.231 |
Typology | Profile | ||
---|---|---|---|
Fully rectangular | R 50 × 20 | 4.21 | 1000 |
U | U 60 × 30 | 2.91 | 646 |
Double T | H 100 × 50 × 4 × 6 | 2.12 | 952 |
Hollow rectangular | RHS 120 × 40 × 4 | 1.47 | 1216 |
Hollow rectangular | RHS 100 × 50 × 4 | 1.19 | 1136 |
Summary of Analytical Approach | |||
---|---|---|---|
Buckling Point | Plasticization Point | ||
Lower-Bound Curve | Upper-Bound Curve | ||
Single X-CBF | |||
Mono-floor building | |||
Multi-storey building |
2.50 | kN/m2 | |
1.20 | kN/m2 | |
1.33 | kN/m2 |
Element | Profile |
---|---|
Beam 1 | IPE 400 |
Beam 2 | IPE 330 |
Beam 3 | IPE 300 |
Beam 4 | IPE 240 |
Column | HEB 200 |
Diagonal − q = 1 | RHS 10 × 6 × 8 |
Diagonal − q = 4 | RHS 8 × 4 × 2.5 |
3.00 | kN/m2 | |
3.00 | kN/m2 | |
1.20 | kN/m2 | |
1.33 | kN/m2 |
Element | Profile |
---|---|
Beam 1 | IPE 450 |
Beam 2 | IPE 360 |
Beam 3 | IPE 300 |
Beam 4 | IPE 240 |
Column | HEB 240 |
Diagonal Profiles − q = 1 | ||
---|---|---|
Level 1: RHS 300 × 50 × 8 | 5344 mm2 | |
Level 2: RHS 250 × 50 × 8.5 | 4811 mm2 | |
Level 3: RHS 250 × 50 × 6.5 | 3731 mm2 | |
Level 4: RHS 180 × 40 × 4.5 | 1899 mm2 |
Diagonal Profiles − q = 4 | ||
---|---|---|
Level 1: RHS 100 × 60 × 3 | 914 mm2 | |
Level 2: RHS 100 × 50 × 3 | 854 mm2 | |
Level 3: RHS 80 × 40 × 3 | 674 mm2 | |
Level 4: RHS 60 × 40 × 2 | 374 mm2 |
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Amadio, C.; Bomben, L.; Noè, S. Analytical Pushover Curves for X-Concentric Braced Steel Frames. Buildings 2022, 12, 413. https://doi.org/10.3390/buildings12040413
Amadio C, Bomben L, Noè S. Analytical Pushover Curves for X-Concentric Braced Steel Frames. Buildings. 2022; 12(4):413. https://doi.org/10.3390/buildings12040413
Chicago/Turabian StyleAmadio, Claudio, Luca Bomben, and Salvatore Noè. 2022. "Analytical Pushover Curves for X-Concentric Braced Steel Frames" Buildings 12, no. 4: 413. https://doi.org/10.3390/buildings12040413