Using System Reliability Concepts to Derive Partial Safety Factors for Punching Shear with Shear Reinforcement: An Explorative Analysis
Abstract
:1. Introduction
- (1)
- Failure inside the zone of punching shear reinforcement (i.e., failure in the surface crossing one or more rows of punching shear reinforcement);
- (2)
- Failure outside the zone of punching shear reinforcement (i.e., failure in the surface located in the region outside the punching shear reinforcement elements);
- (3)
- Failure at the maximum punching shear capacity (i.e., failure in the compression zone directly at the face of the column).
2. Designing Flat Slabs against Punching Shear Failures with Shear Reinforcement
2.1. General Considerations
2.2. Design Provisions for Punching Shear with Shear Reinforcement according to DIN EN 1992-1-1 (Eurocode 2)
2.3. Complementary Design Provisions for Punching Shear with Shear Reinforcement in the German National Annex NA(D)
3. Foundations of the Theory of Structural Systems Reliability
4. Considerations for the Reliability Analyses of the Structural System
4.1. Database of Experimental Tests and Data Filtering Criteria
- (1)
- The tests should contain at least one row of punching shear reinforcement.
- (2)
- Only star-like punching shear reinforcement was used.
- (3)
- Only stirrup reinforcement (or comparable reinforcement) was considered (note that this criterion does not apply to failure mode (ii) referring to failure outside the punching shear reinforcement zone).
- (4)
4.2. Stochastic Models for the Basic Variables
Concrete Class | Prior Parameters (*) [38] | Parameters for a Lognormal Distribution | fcm | fck | |||||
---|---|---|---|---|---|---|---|---|---|
Standard | |||||||||
C25 | 3.65 | 3.0 | 0.12 | 10 | 3.65 | 0.164 | 39.00 | 6.45 | 28.36 |
C35 | 3.85 | 3.0 | 0.09 | 10 | 3.85 | 0.123 | 47.35 | 5.86 | 38.37 |
C45 | 3.98 | 3.0 | 0.07 | 10 | 3.98 | 0.096 | 53.76 | 5.17 | 45.71 |
Failure Mode (Design Equation) | Parameter | Normal Distribution (ND) | Lognormal Distribution (LND) | |||||
---|---|---|---|---|---|---|---|---|
Point Estimators | Interval Estimators (IE) (95%) | Point Estimators | Interval Estimators (95%) | |||||
LND Values | LND Values Converted | IE Values | IE Values Converted | Corrected Values | ||||
(i) Inside shear reinf. zone (Equation (4)) | μ | 1.3927 | [1.3397; +∞] | 0.3219 | 1.3927 (a) | [0.2841; +∞] | [1.3478; +∞] (d) | 1.3478 |
σ | 0.1910 | [–∞; 0.2375] | 0.1362 | 0.1906 (b) | [–∞; 0.1694] | [–∞; 0.2300] (e) | 0.2199 (g) | |
CoV | 0.1371 | [–∞; 0.1773] | 0.4231 | 0.1368 (c) | [–∞; 0.5962] | [–∞; 0.1706] (f) | 0.1631 | |
(ii) Outside shear reinf. zone (Equation (10)) | μ | 1.1072 | [1.0654; +∞] | 0.0903 | 1.1075 (a) | [0.0514; +∞] | [1.0713; +∞] (d) | 1.0713 |
σ | 0.1648 | [–∞; 0.2008] | 0.1536 | 0.1712 (b) | [–∞; 0.1871] | [–∞; 0.2023] (e) | 0.1951 (g) | |
CoV | 0.1489 | [–∞; 0.1885] | 1.7013 | 0.1545 (c) | [–∞; 3.6442] | [–∞; 0.1888] (f) | 0.1821 | |
(iii) At max. punching shear capacity (Equation (9)) | μ | 1.0855 | [1.0567; +∞] | 0.0769 | 1.0855 (a) | [0.0500; +∞] | [1.0596; +∞] (d) | 1.0596 |
σ | 0.1095 | [–∞; 0.1345] | 0.1021 | 0.1111 (b) | [–∞; 0.1254] | [–∞; 0.1334] (e) | 0.1224 (g) | |
CoV | 0.1009 | [–∞; 0.1273] | 1.3277 | 0.1023 (c) | [–∞; 2.5058] | [–∞; 0.1259] (f) | 0.1155 |
Basic Variables | Distr. | Unit | Nominal Value | Mean | Stand. | |
---|---|---|---|---|---|---|
Concrete compressive strength fc (also see Table 1) | C25 | LSD | N/mm² | 25 | 39.00 | 6.45 |
C35 | 35 | 47.35 | 5.86 | |||
C45 | 45 | 53.76 | 5.17 | |||
Yield strength | fy | ND | N/mm² | 500 | 30 | |
Effective depth | d | ND | mm | dnom | dnom +10 | 10 |
Column dimensions | c, c∅ | ND | mm | cnom | cnom + 0.003cnom | 4 + 0.006cnom |
Slab width | h | ND | mm | hnom | hnom + 10 | 10 |
Reinf. area | As | ND | mm² | As,nom | As,nom | 0.02As,nom |
Radial spacing of perimeter of shear reinforcement | sr | ND | mm | sr,nom | sr,nom | sr,nom ± 20 |
Model uncertainty: | ||||||
(i) Inside reinf. zone | LND | - | 1.3478 | CoV: 0.1631 | ||
(ii) Outside reinf. zone | LND | - | 1.0713 | CoV: 0.1821 | ||
(iii) At max. punching shear capacity | LND | - | 1.0596 | CoV: 0.1155 |
4.3. Computational Considerations
4.3.1. Design of the Structural System
4.3.2. Reliability-Based Methods (Levels II and III Methods)
5. Results
5.1. System Reliability Analyses for Fixed Partial Factors
5.2. System Reliability Analyses for Variable Partial Factors (Sensitivity Analysis)
6. Discussion
7. Conclusions
- The concepts related to systems reliability can be applied to derive partial factors for design equations. The above-described results indicated that the partial factors can vary considerably depending on the governing limit state function and the scatter of the basic variables. This confirmed the importance of a thorough definition of stochastic models for the basic variables and the importance of a careful selection of the reliability-based method (i.e., level II or level III method).
- In this investigation, it was seen that the use of the simple bounds in combination with a first-order reliability method (i.e., level II methods) can give a decent preliminary indication of the system safety level. Yet, the results should be carefully interpreted to avoid deriving unsafe partial factors. If sufficient computing power is available, simulation methods (i.e., level III methods) should be used since they enable the calculation of more precise system failure probabilities.
- The results of the sensitivity analysis confirmed the premise that a thorough analysis of the reliability may support the reduction of partial factors . This could be a possible approach for practitioners to save material and costs and reduce CO2 emissions during the structural design process without compromising socially accepted safety levels (e.g., the value of 3.8 recommended in DIN EN 1990 [22]).
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Design Parameters | Values | |
---|---|---|
Concrete compressive strength fck | 35 N/mm² | |
Yield strength fy | 500 N/mm² | |
Effective depth d | 280 mm | |
Column dimensions c∅ | 500 mm | |
Slab width h | 320 mm | |
Cross-sectional reinforcement area of the first and second rows | Varying between 1334 mm² (min.) and 6000 mm² (max.) | |
Flexural reinforcement ratio (*) | 0.8% | |
5.61 |
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Feiri, T.; Schulze-Ardey, J.P.; Ricker, M.; Hegger, J. Using System Reliability Concepts to Derive Partial Safety Factors for Punching Shear with Shear Reinforcement: An Explorative Analysis. Appl. Sci. 2023, 13, 1360. https://doi.org/10.3390/app13031360
Feiri T, Schulze-Ardey JP, Ricker M, Hegger J. Using System Reliability Concepts to Derive Partial Safety Factors for Punching Shear with Shear Reinforcement: An Explorative Analysis. Applied Sciences. 2023; 13(3):1360. https://doi.org/10.3390/app13031360
Chicago/Turabian StyleFeiri, Tânia, Jan Philip Schulze-Ardey, Marcus Ricker, and Josef Hegger. 2023. "Using System Reliability Concepts to Derive Partial Safety Factors for Punching Shear with Shear Reinforcement: An Explorative Analysis" Applied Sciences 13, no. 3: 1360. https://doi.org/10.3390/app13031360