# Probabilistic and Semi-Probabilistic Analysis of Slender Columns Frequently Used in Structural Engineering

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## Abstract

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## 1. Introduction

- (a)
- The clear definition of the steps required for the probabilistic verification of the N-M stability of slender columns with regard to scattering interaction diagrams (I-D) [see EN 1992-1] [8].
- (b)
- (c)
- The study of the sensitivities, which vary with the load level (N-M interaction load level), of the descriptive model input variables for both the column cross-sectional level and the column component level in relation to the column load-bearing capacity and the column deformation.
- (d)
- The determination of the global safety resistance factors according to the “Estimation of the Co-efficient of Variation” (ECOV) method using the non-linear finite element responses generated by means of Latin Hyper Cube Sampling and from responses to experiments as well as the suitability of the global safety resistance factors for reliability assessment.

## 2. Probabilistic Non-Linear Computation

#### 2.1. Reliability Levels

_{i}(i = 1, …, m). For the present structural reliability problem, g(x) is formulated as the difference between the resistance R and the load E:

_{R}for the resistance and K

_{E}for the load are introduced [23], as they are meant to reduce the deviation of the numerical model from the realistic model. The limit state function is formulated in a way that negative values indicate failure and the failure probability is defined as the probability that the random combination of the input values results in an outcome in the failure domain. Mathematically the latter can be expressed through Equation (3)

_{x}(.) is the m-dimensional joint probability density function (PDF) of the m basic variables X

_{i}. The structural reliability is quantified through the reliability index β, which can be generally expressed and calculated—assuming a normal distribution for g(x)—through Equation (4).

^{−1}[.] is the inverse of the standard normal cumulative distribution function.

- Level III: limit state functions and distribution functions for the random variables are introduced without any approximation; calculations are usually based on Monte Carlo simulation or straightforward numerical integration;
- Level II: the amount of calculation efforts is reduced by adopting well-chosen linearization techniques, usually the so called First Order Reliability Method; the degree of accuracy may strongly depend on the details of the problem at hand;
- Level I: the variables Xi are introduced by one single value only; this value is referred to as the design value. This method does not actually calculate a failure probability but only checks whether some defined target level is attained or not. It is the basis for partial safety factor format (PSFM) which is defined in Eurocodes as the basic design format for new structures and it is the design and assessment procedure in everyday practice and is referred to as the semi-probabilistic level.

- It is assumed that the variables in a limit state function are in a first approach independent from each other. Although correlations between variables can be taken into account in computational programs, they are difficult to determine and convolute the algorithms.
- The analysis does not take into account human error. The failure probabilities p
_{f}discussed herein are conditional on the assumption that there are no errors affecting the resistance and loading condition of the case study. To reduce errors, special strategies and quality control measures are required.

#### 2.2. Limit State Design

**,**verifications must be performed in order to obtain an adequate level of resistance against the loads imposed on the structures. The MC2010 proposes (a) the probabilistic method, (b) global resistance factor format (GRF) and (c) the partial safety factor format (PF). The Estimate of Coefficient of Variation of Resistance (ECOV) method is a particular method within the global resistance format. The forthcoming fib Model Code 2020, although its content and formats are still under discussion, should also encompass this concept.

_{m}is the mean resistance value and R

_{k}is the characteristic value of resistance corresponding to 5% exceedance probability. The respective global safety factor ${\gamma}_{R}$ is calculated as:

_{R}, the coefficient of variation V

_{R}of the resistance side and the reliability index ß are considered. The descriptive elements of the action side (e.g., partial safety factor γ

_{E}) influences the consideration only indirectly via the interrelation between α

_{E}, α

_{R}and ß.

#### 2.3. Sampling Methods

_{ik}of the underlying distributions for the random variables X

_{i}where the index “k” stands for the “k”-th simulation (k = 1, 2, …, N

_{real}), see Figure 1. When combined with a finite element model, the resistance of the analyzed component or the limit state function has to be calculated for a large number of repetitions, yielding an output distribution function. Each set of the k realizations introduced into the analyzed model leads to a solution.

_{k}are evaluated statistically according to the basic statistics and lead to the p

_{f}= z

_{0}/z, where z

_{0}is the number of results violating the design threshold.

_{sim}, where N

_{sim}= amount of simulations; p

_{i}is generated randomly with uniform distribution on the interval $\left[0-1\right]$. However, sample points are generated without considering the previously generated sample points.

_{i}is chosen randomly when using the MC method, see Figure 1 left, one p

_{i}is chosen from each of the N

_{sim}-intervals, see Figure 2 left, when using the LHS method. Random samples can be taken one at a time, remembering which samples were taken so far by dividing the cumulative density function (CDF) into N

_{sim}equally probable k intervals, see Figure 2. It ensures an acceptable accuracy at a low N

_{sim}.

_{sim}-intervals but taking only a reduced number of p

_{i}s from this sample field for the evaluation procedures into account. The selection of the reduced p

_{i}s is based on predefined fractile values of the CDF, for instance as shown in Figure 3. Hence, the effort of simulations can be reduced while fully including the properties of the LHS sampling field—encompassing the N

_{sim}-intervals.

_{i}s to predefined fractile values is only possible for one basic variable, it is necessary to define a leading basic variable in advance and to formulate the dependencies of the other basic variables on this via the LHS simulation field.

_{L}; (d) determine those realizations which are closest to the pre-defined fractiles (e.g., 5%, 15%, …, 95%); (e) select the sample sets associated with the pre-defined fractiles of the leading parameter X

_{L}; (f) perform the deterministic simulation of each subset; (g) collect the results of the system response X

_{R}and order it according to the pre-defined fractiles; (h) perform a PDF-fitting; (i) perform sensitivity analysis to (I) the original LHS-field ${\rho}_{{\alpha}_{LHS}}$ and (II) the reduced FBS-field ${\rho}_{{\alpha}_{FBS}}$; (f) check leading parameter $|{\rho}_{{\alpha}_{LHS}}-{\rho}_{{\alpha}_{FBS}}|\le error$.

_{i}s as it is proposed in the FBS, but it is a simplified probabilistic procedure in which the random variation of resistance is estimated using only two samples. It is based on the idea that the random distribution of resistance, which is described by the COV, can be estimated from the mean and characteristic values e.g., 5%-fractile of resistance, see Figure 4. The method is not based on an LHS sampling field and hence does not take into account the correlation between basic variables but takes from all basic variables the, e.g., 5%-fractile or 50%-fractile.

## 3. Investigated Columns

#### 3.1. Layout and Test Results

_{max}of each single column. All of the maximum N-M gradient points show the system stability failure before the mean I-D (left from the I-D

_{m}) as defined in EN 1992-1-1 [8]. The right-hand graphic of Figure 6 shows the load-vs.-strain graphs in the fracture-prone cross-section at half the height of the columns. The concrete compressive strains in the compressed fiber of the cross-section were recorded for the column stability loss at between 1.4 and 1.8‰ and were far away from the permissible concrete compressive strains of 3.5‰. The associated concrete/reinforcement tension strains in the pulled fiber of the cross-section were recorded at between 1.4 and 3.1‰, see Figure 6. The model uncertainties of the experimental test results were derived following [30] and EN1990 Annex D (Edition: 2013-03-15) with θ

_{Y}= Y

_{mean}/Y

_{k}. This results for N

_{max}to θ

_{Nmax}= 1.06, for e

_{2}at N

_{max}to θ

_{e}

_{2,Nmax}= 1.02, and for M at N

_{max}to θ

_{M}

_{,Nmax}= 1.01.

#### 3.2. Reliability Assessment

_{f}= 10

^{−6}is defined as the minimum requirement for the bearing capacity of structures.

## 4. Probabilistic Analyses

_{R}is the model uncertainty related to the resistance, R denotes the resistance of the respective scenario in terms of ultimate load bearing capacity, K

_{E}is the model uncertainty related to the loads, represents the permanent loads acting on the structure while q represents the imposed service loads. In the following, the probabilistic models of the different variables in these equations are discussed.

_{R}· R) in Equation (9) corresponds to the interaction diagram (I-D) which corresponds to the function of the maximum permissible N-M values, while the action side (K

_{E}·(g + q)) of Equation (9) corresponds to the acting N-M load path so that the intersection of the N-M load path with the I-D characterizes the maximum permissible N-M values; further details can also be found in [33,34]. Both strategies have it in common that the model uncertainties are taken into account in determining the necessary partial safety factors.

#### 4.1. Slenderness

- $i$ is the minimum radius of gyration: $i=\sqrt{{I}_{c}/{A}_{c}}$
- ${I}_{c}$ is the moment of inertia
- ${A}_{c}$ is the concrete cross-sectional area
- ${l}_{o}$ is the effective length of the member which can be assumed to be:$${l}_{o}=\beta \xb7{l}_{w}$$

- ${l}_{w}$ is the clear height of the member
- β is a coefficient which depends on the support conditions.

_{w}is included in the slenderness evaluation via the equivalent length l

_{o}. The procedures for determining the equivalent length (nomograms) have been adopted in Eurocode 2 along with the routines for determining the ß value. In addition, the verification of the load-bearing capacity is carried out in the critical cross-section in the following outlines of simplified procedures.

#### 4.2. Simplified Design Formats

- ${N}_{\mathrm{Rd}}$ is the axial resistance
- $b$ is the overall width of the cross-section
- ${h}_{\mathrm{w}}$ is the overall depth of the cross-section
- f
_{cd,pl}is the design compressive strength for plain concrete

_{cc,pl}= 0.8

- ${e}_{0}$ is the first order eccentricity including, where relevant, the effects of floors (e.g., possible clamping moments transmitted to the wall from a slab) and horizontal actions.
- ${e}_{i}$ is the additional eccentricity covering the effects of geometrical imperfections.

_{o}, see Section 4.1, (b) the slenderness λ, (c) the load center of the action e

_{o}= M

_{sd}/N

_{sd}, (d) the unwanted eccentricity e

_{a}, (e) e

_{2}(theory II order effects), (f) e

_{tot}= e

_{o}+ e

_{a}+ e

_{2}and (g) dimensioning for N

_{sd}and M

_{sd}= N

_{sd}·e

_{tot}using a µ–ν diagram or Equations (12)–(14).

#### 4.3. Non-Linear Analysis Formats

#### 4.4. Simplified Basic Variables X_{i}

#### 4.5. Elements of the Limit State Formulations

_{Zug}is the section height under tension. For the control as to whether predominantly bending failure takes place can be tracked by:

_{1}to X

_{11}. For these randomly generated values, the cross-sectional resistance was then calculated (see Table 3, bottom part) allowing the creation of a point cloud around the mean I-D as shown in figure in Section 4.7.

#### 4.6. Sensitivity Analyses

#### 4.6.1. Cross-Sectional Level

_{i}are used to provide the relative importance of each individual random variable. By definition, the sum of squares of sensitivity factors for each random variable is equal to 1:

_{1}(E

_{ci}) has a major impact on results but as the force increases, its impact diminishes as the concrete is entering a non-linear state of behavior. Due to small uncertainties in the statistical parameters of areas of reinforcement, their impact on the results can be neglected. The axis distance of reinforcement in compression X

_{7}(d

_{1}) has a major impact at the beginning of the loading, but reduces as the force increases, while the impact of axis distance of reinforcement in tension X

_{8}(d

_{2}) increases with the force and deformation of the concrete in tension. Its impact reaches its peak values as the concrete deformation in tension reaches its maximum, just before the cracking of the concrete. The impact of the concrete compressive strength X

_{11}(f

_{c}) increases along with the loading and reaches its peak value at the point when the concrete deformation in compression and tension are equal (absolute values). After the peak (when the force is around 280 kN), its impact decreases as the tension area of the concrete increases.

#### 4.6.2. Component Level

_{7}and X

_{8}were not taken into account in the sensitivity analyses because (a) the exact locations of the reinforcement were quarantined by an extraordinary quality control during the fabrication of the columns and (b) the influence of the material laws, solution algorithms, non-linear fracture processes and slenderness on the instability process were the focus of interest. As can be seen in Figure 9a, the modulus of elasticity of the concrete X

_{1}(E

_{ci}) as well as the compressive strength X

_{11}(f

_{c}) dominate in the ultimate failure load in compression. When considering the sensitivity factors with regard to the horizontal deflection in the middle height of the column, see Figure 9b, the tensile strength of the concrete X

_{12}(f

_{ct}) and the compressive strength X

_{11}(f

_{c}) both play a significant role initially; however, the modulus of elasticity of the concrete X

_{1}(E

_{ci}) becomes more important with increasing horizontal deflection. On the basis of the 60 Latin Hyper Cube samples, Figure 9d provides an insight into the scatters in the relationship between the axial force and the horizontal displacement in the middle height of the column, computed in ATENA.

_{1}(E

_{ci}), as well as the compressive strength X

_{11}(f

_{c}) and fracture energy X

_{13}(G

_{f}) dominate in the ultimate failure load. When considering the sensitivity factors with regard to the horizontal deflection in the column middle height, see Figure 10b, the tensile strength of the concrete X

_{12}(f

_{ct}) and the compressive strength X

_{11}(f

_{c}) both play a significant role as the horizontal displacement increases, however, the modulus of elasticity of the concrete X

_{1}(E

_{ci}) and the concrete fracture energy X

_{13}(G

_{f}) are initially more important. Figure 10d provides an insight into the scatters in the relationship between the axial force and the horizontal displacement in the middle height of the column made of C100/115 in a similar way as was reported in the previous section for the C45/55 column.

_{1}(E

_{ci}) becomes more important for the normal force analysis with higher strength, see Figure 9a and Figure 10a. For the horizontal deflection sensitivity analysis, X

_{1}(E

_{ci}) becomes less important and X

_{13}(G

_{f}) becomes more important with increasing strength and load, see Figure 9b and Figure 10b.

#### 4.7. Standard Based Analyses

_{i}for the limit state equation. For the correlation between the input parameters of the EN1992-1-1 [8], provision was not explicitly discussed, since the provision formulation implements correlations implicitly. For the numerical probabilistic analysis, an explicit definition of the correlations between the input parameters was made according to [31].

#### 4.8. Non-Linear Finite Element Analyses

#### 4.8.1. General

_{i}for the limit state equation, (e) the generation of the n-simulation sets, or sample sets using LHS technique for non-linear finite element calculations (for LHS see Section 2.3, for NLFEM see Section 4.3) based on the probabilistic parameters of the random variables displayed in Table 4 and Table 5, (f) the n-fold repetition of the NLFEM computation and the structuring of the statistical system responses in response vectors of dimension n suitable for the reliability assessment and (g) the determination of the reliability and failure probabilities with regard to predefined limit values and the study of the sensitivities of the input base variables with regard to the examined limit state equations.

#### 4.8.2. Pre NLFEM-Modeling

#### 4.8.3. Probabilistic NLFEM-Modeling

_{m}on the mean level, see Figure 12b on the right. Such an underestimation of moment capacity was not seen in the simulations of the partners UNIZG-FCE and U-MINHO, see Figure 12c,d. Therefore, we can conclude that BOKU, UNIZG-FCE and U-MINHO came to completely different results for moment distribution. The probabilistic analyses of most of the partners showed that the calculated normal force mean values are significantly higher than the experimentally determined ones, as can be seen from the box whisker plots (P-NLFEM results in pink; experimental results in blue) and the histograms of Figure 12. In these histograms of the normal forces it can also be seen that these differences are smaller in the lower fractile ranges.

_{mean}according to EN1992-1 indicate a possible defect and possibly safety problems in the I-D formulations too, in particular for the examined columns geometry.

#### 4.9. ECOV Analyses

_{Rd}= 1.1). For low uncertainty models (γ

_{Rd}= 1.06), the overestimation is observed for the U-MINHO and BOKU NLFEM. As previously stated, the nominal stiffness method provides results that are too conservative, with a design normal force representing approximately 70% of the design normal force provided by the experimental campaign results.

_{Rd}>1.1) are recommended for the design and safety assessment of slender columns using standard NLFEM. Further research is also suggested to investigate more sophisticated numerical models for the prediction of the carrying capacity of slender columns.

## 5. Conclusions

_{Rd}> 1.10) are recommended for the design and safety assessment of slender columns using standard NLFEM.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- JCSS. JCSS Probabilistic Model Code. Available online: https://www.jcss-lc.org/jcss-probabilistic-model-code/ (accessed on 18 August 2021).
- Faber, M. Statistics and Probability Theory; Springer: Berlin, Germany, 2012. [Google Scholar]
- Benko, V. Nichtlineare Berechnung von Stahlbetondruckglieder. (Nonlinear analysis of reinforced concrete compression members). Innov. Betonbau
**2001**, 27, 9–12. [Google Scholar] - Strauss, A.; Ivanković, A.M.; Benko, V.; Matos, J.; Marchand, P.; Wan-Wendner, R.; Galvão, N.; Orcesi, A.; Dobrý, J.; Diab, M.E.H.; et al. Round-Robin Modelling of the Load-bearing Capacity of Slender Columns by Using Classical and Advanced Non-linear Numerical and Analytical Prediction Tools. Struct. Eng. Int.
**2021**, 31, 118–135. [Google Scholar] [CrossRef] - Benko, V.; Gúcky, T.; Valašík, A. The reliability of slender concrete columns subjected to a loss of stability. In Advances and Trends in Engineering Sciences and Technologies II, Proceedings of the 2nd International Conference on Engineering Sciences and Technologies, ESaT 2016, Vysoké Tatry, Slovak Republic, 29 June–1 July 2016; Taylor & Francis: London, UK, 2017. [Google Scholar] [CrossRef]
- Benko, V.; Dobrý, J.; Čuhák, M. Failure of Slender Concrete Columns Due to a Loss of Stability. Slovak J. Civ. Eng.
**2019**, 27, 45–51. [Google Scholar] [CrossRef][Green Version] - CEB-FIP. Practitioners’ Guide to Finite Element Modelling of Reinforced Concrete Structures; Fib Fédération Internationale du Béton: Lausanne, Switzerland, 2008. [Google Scholar]
- EN 1992-1-1. Eurocode 2: Design of Concrete Structures—Part 1-1: General Rules and Rules for Buildings; European Standard: Brussels, Belgium, 2004; Volume 1. [Google Scholar]
- Shlune, H.; Gylltoft, K.; Plos, M. Safety format for non-linear analysis of concrete structures. Mag. Concr. Res.
**2012**, 64, 563–574. [Google Scholar] [CrossRef] - Holicky, M. Global resistance factors for reinforced concrete members. In Proceedings of the 1st International Symposium on Uncertainty Modelling in Engineering, Prague, Czech Republic, 2–3 May 2011. [Google Scholar]
- Cervenka, V. Global Safety formats in Fib Model Code 2010 for Design of Concrete Structures. In Proceedings of the 11th Probabilistic Workshop, Brno, Czech Republic, 6–8 November 2013. [Google Scholar]
- Cervenka, V. Reliability-based non-linear analysis according to Model Code 2010. Struct. Concr.
**2013**, 14, 19–28. [Google Scholar] [CrossRef] - Caspeele, R.; Steenbergen, R.; Sykora, M. Partial Factor Methods for Existing Concrete Structures; FIB Bulletin No. 80; FIB: Lausanne, Switzerland, 2016; ISBN 978-2-88394-120-5. [Google Scholar] [CrossRef]
- Engen, M.; Hendriks, M.; Köhler, J.; Øverli, J.; Åldtstedt, E. A quantification of modelling uncertainty for non-linear finite element analysis of large concrete structures. Struct. Saf.
**2017**, 64, 1–8. [Google Scholar] [CrossRef] - Castaldo, P.; Gino, D.; Bertagnoli, G.; Mancini, G. Partial safety factor for resistance model uncertainties in 2D non-linear analysis of reinforced concrete structures. Eng. Struct.
**2018**, 176, 746–762. [Google Scholar] [CrossRef] - Moccia, F.; Yu, Q.; Ruiz, M.F.; Muttoni, A. Concrete compressive strength: From material characterization to a structural value. Struct. Concr.
**2021**, 22, E655–E682. [Google Scholar] [CrossRef] - Momeni, M.; Bedon, C. Uncertainty Assessment for the Buckling Analysis of Glass Columns with Random Parameters. Int. J. Struct. Glass Adv. Mater. Res.
**2020**, 4, 254–275. [Google Scholar] - Mehmel, A.; Schwarz, H.; Karperek, K.; Makovi, J. Tragverhalten Ausmittig Beanspruchter Stahlbetondruckglieder; Institut Für Baustatik, EHT, Deutscher Ausschuss für Stahlbeton, Heft. 204; DafStb: Berlin, Germany, 1969. [Google Scholar]
- Foster, S.; Attard, M. Experimental tests on eccentrically loaded high strength concrete columns. Struct. J.
**1997**, 94, 295–303. [Google Scholar] - Allaix, D.; Carbone, V.; Mancini, G. Global safety format for non-linear analysis of reinforced concrete structures. Struct. Concr.
**2013**, 14, 29–42. [Google Scholar] [CrossRef] - Cervenka, V.; Cervenka, J.; Kadlek, L. Model uncertainties in numerical simulations of reinforced concrete structures. Struct. Concr.
**2018**, 19, 2004–2016. [Google Scholar] [CrossRef] - Gino, D.; Castaldo, P.; Giordano, L.; Mancini, G. Model uncertainty in non-linear numerical analyses of slender reinforced concrete members. Struct. Concr.
**2021**, 22, 845–870. [Google Scholar] [CrossRef] - Ditlevsen, O.; Madsen, H. Structural Reliability Methods; John Wiley & Sons Ltd.: Chicheste, Denmark, 1996. [Google Scholar]
- Ang, A.H.; Tang, W.H. Probability Concepts in Engineering Planning, 2nd ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2007. [Google Scholar]
- Hendriks, M.A.N.; de Boer, A.; Belletti, B. Guidelines for Nonlinear Finite Element Analysis of Concrete Structures: Girder Members; Report RTD:1016:2012; Rijkswaterstaat Ministry of Infrastructure and Water Management: The Hague, The Netherlands, 2012. [Google Scholar]
- European Committee for Standardization (CEN). EN 1990, Eurocode 0: Basis of Structural Design; CEN: European Standard: Brussels, Belgium, 2005. [Google Scholar]
- International Federation for Structural Concrete (FIB). Model Code for Concrete Structures; Ernst & Sohn: Lausanne, Switzerland, 2010. [Google Scholar]
- Novák, D.; Vořechovský, M.; Teplý, B. FReET: Software for the statistical and reliability analysis of engineering problems and FReET-D: Degradation module. Adv. Eng. Softw. Adv. Eng. Softw.
**2014**, 72, 179–192. [Google Scholar] [CrossRef] - Beletti, B.; Vecchi, F.; Cosma, M.P.; Strauss, A. Non linear structural analyses of prestressed concrete girder: Tools and safety formats. In Life-Cycle Analysis and Assessment in Civil Engineering. Towards an Integrated Vision, Proceedings of the Sixth International Symposium on Life-Cycle Civil Engineering, Ghent, Belgium, 28–31 October 2008; Caspeele, R., Taerwe, L., Frangopol, D.M., Eds.; Taylor & Francis Group: London, UK, 2018. [Google Scholar]
- Achenbach, M.; Gernay, T.; Morgenthal, G. Quantification of model uncertainties for reinforced concrete columns subjected to fire. Fire Saf. J.
**2019**, 108, 102832. [Google Scholar] [CrossRef] - Zimmermann, T.; Lehký, D.; Strauss, A. Correlation among selected fracture-mechanical parameters of concrete obtained from experiments and inverse analyses. Struct. Concr.
**2016**, 17, 1094–1103. [Google Scholar] [CrossRef] - Strauss, A.; Krug, B.; Slowik, O.; Novak, D. Combined shear and flexure performance of prestressing concrete T-shaped beams: Experiment and deterministic modeling. Struct. Concr.
**2018**, 19, 16–35. [Google Scholar] [CrossRef][Green Version] - Strauss, A.; Benko, V.; Taubling, B.; Valasik, A.; Cuhak, M. Reliability of slender columns. Beton-Stahlbetonbau
**2017**, 112, 392–401. [Google Scholar] [CrossRef] - Červenka, V.; Jendele, L.; Červenka, J. Atena Program Documentation—Part 1: Theory; Cervenka Consulting: Prague, Czech Republic, 2007. [Google Scholar]
- Tau, K.; Agresti, A. Analysis of Ordinal Categorical Data, 2nd ed.; John Wiley & Sons: New York, NY, USA, 2010. [Google Scholar]
- Strauss, A.; Novák, D.; Lehký, D.; Vořechovský, M.; Teplý, B.; Pukl, R.; Červenka, V.; Eichinger-Vill, E.M.; Santa, U. Safety analysis and reliability assessment of engineering structures—The success sotry of SARA. Ce/Papers
**2019**, 3, 38–47. [Google Scholar] [CrossRef] - Strauss, A.; Wan-Wendner, R.; Vidovic, A.; Zambon, I.; Yu, Q.; Frangopol, D.M.; Bergmeister, K. Gamma prediction models for long-term creep deformations of prestressed concrete bridges. J. Civ. Eng. Manag.
**2017**, 23, 681–698. [Google Scholar] [CrossRef]

**Figure 1.**Monte-Carlo method simulation procedure including and not including correlation between basic variables.

**Figure 2.**Latin Hyper Cube Simulation (LHS) method including and not including correlation between basic variables.

**Figure 3.**Fractile Based Simulation (FBS) method procedure including and not including correlation between basic variables—including the process steps.

**Figure 6.**Experimentally determined N-M gradients vs. the Eurocode I-D (blue graph = design values, red dashed graph = characteristic values, red graph = mean values) (

**a**) and normal force—strain (

**b**) of the specimens S1-1 to S1-6.

**Figure 8.**Sensitivity analyses at the cross-section level: (

**a**) sensitivity of material parameters with respect to the applied normalized normal force (maximum bearing capacity of N = 335.5 kN); (

**b**) strains in compression and tension with respect to the applied normal force, see also Table 3.

**Figure 9.**Sensitivity analyses at the component/column level based on the probabilistic non-linear analysis of the considered column made of C45/55, see Section 4.4: (

**a**) sensitivity of material parameters with respect to the normalized applied axial force; (

**b**) sensitivity of material parameters with respect to the normalized horizontal displacement at middle height of the column; (

**c**) basic variables of considered material models; (

**d**) normalized horizontal displacement vs. normalized axial force.

**Figure 10.**Sensitivity analyses at the component/column level based on the probabilistic non-linear analysis of the considered column made of C100/115, see Section 4.4: (

**a**) sensitivity of material parameters with respect to the normalized applied axial force; (

**b**) sensitivity of material parameters with respect to the normalized horizontal displacement at middle height of the column; (

**c**) basic variables of considered material models; (

**d**) normalized horizontal displacement vs. normalized axial force response.

**Figure 11.**Statistical scattering failure loads obtained according to EN1992-1 nominal stiffness calculations (pink point set). (

**a**) Nominal stiffness calculated and experimental N-M gradients vs. the Interaction Diagrams I-D according to EN1992-1 at the design, characteristic and mean levels, and, (

**b**) histogram of the maximum calculated axial forces (pink) vs. those tested experimentally (blue) and histogram of the maximum calculated moments (pink) vs. those tested experimentally (blue).

**Figure 12.**Statistical structural responses and PDFs obtained according to EN1992-1 non-linear finite element analyses: (

**a**) BOKU Group: University of Natural Resources and Life Sciences Vienna, Institute of Structural Engineering, (

**b**) STUBA Group: Slovak University of Technology in Bratislava, Department of Concrete Structures and Bridges, (

**c**) U-MINHO Group: University of Minho, Institute for Sustainability and Innovation in Structural Engineering (ISISE) and (

**d**) UNIZG-FCE Group: University of Zagreb Faculty of Civil Engineering, Department of Structural Engineering.

Description | Symbol | Unit | Value |
---|---|---|---|

Width of column cross-section | b | mm | 240 |

Height of column cross-section | h | mm | 150 |

Distance from topmost compression face to the centroid of compression reinforcement | d’ | mm | 33 |

Distance from topmost compression face to the centroid of tensile reinforcement | d | mm | 117 |

Cross-sectional area of RC column | A_{c} | mm^{2} | 36,000 |

Cross-sectional area of tensile reinforcement | A_{s} | mm^{2} | 307.88 |

Cross-sectional area of compression reinforcement | A_{’s} | mm^{2} | 307.88 |

Characteristic compressive strength of concrete (for assoc. parameters see table in Section 4.5 | f’_{ck} | N/mm^{2} | 55 |

Yield strength of reinforcing reinforcement | f_{sy} | N/mm^{2} | 500 |

Young modulus of reinforcing reinforcement | E_{s} | N/mm^{2} | 200,000 |

Ultimate strain of concrete in compression | ε_{cu} | - | Varied |

Ratio of the depth of equivalent compression block to that of actual compression | β | - | Varied |

Eccentricity | e | mm | 40 |

**Table 2.**Descriptive statistical parameters of the experimental results of the considered test series C45/55 without considering sample size aspects.

Test | N_{max} (kN) | e_{2} (mm) | M_{max} (kNm) |
---|---|---|---|

S1-1 | 324.4 | 57.6 | 31.7 |

S1-2 | 323.4 | 42.7 | 26.8 |

S1-3 | 332.6 | 38.3 | 26.0 |

S1-4 | 271.2 | 58.4 | 26.7 |

S1-5 | 296.0 | 59.4 | 29.4 |

S1-6 | 311.4 | 55.0 | 29.6 |

**Table 3.**Input parameters for the probabilistic analyses of the EN 1992-1-1 closed formulations provisions and the EN 1992-1 Non-Linear Finite Element provisions of the slender column made of C45/55.

X | Variable | Dis. ** | Unit | X_{k} | µ | σ | |
---|---|---|---|---|---|---|---|

C45/55 | |||||||

X_{1} | E_{ci} ^{a,b,c} | Initial tangent concrete modulus of elasticity | LN | GPa | 37.5 | 37.5 | 4.91 |

X_{2} | E_{s} ^{a,b,c} | Reinforcing steel modulus of elasticity | D | GPa | 200 | 200 | - |

X_{3} * | A_{s}_{1}^{b} | Reinforcement area | N | cm^{2} | 3.08 | 3.08 | 0.062 |

X_{4} * | A_{s}_{2}^{b} | Reinforcement area | N | cm^{2} | 3.08 | 3.08 | 0.062 |

X_{5} * | H ^{b} | Height | N | cm | 15.0 | 15.0 | 0.30 |

X_{6} * | B ^{b} | Width | N | cm | 24.0 | 24.0 | 0.45 |

X_{7} * | d ^{b} | Axis distance of reinforcement | LN | cm | 3.30 | 3.30 | 0.50 |

X_{8} * | d_{2}^{b} | Axis distance of reinforcement | LN | cm | 3.30 | 3.30 | 0.50 |

X_{9} * | e_{1}^{b} | Eccentricity | N | cm | 4.00 | 4.00 | 0.10 *** |

X_{10} * | ε_{c,1} ^{b} | Strain at max. compressive stress | LN | ‰ | −2.40 | −2.40 | 0.11 |

X_{11} | f_{c} ^{a,b,c} | Concrete compressive strength | LN | MPa | 45.0 | 53.0 | 5.13 |

X_{12} | f_{ct} ^{a,b,c} | Concrete tensile strength | LN | MPa | 2.7 | 3.8 | 0.78 |

X_{13} | G_{F} ^{a,b,c} | Concrete fracture energy | LN | MPa | 104 | 149 | 30.8 |

X_{14} | ε_{c,lim} ^{a,b,c} | Ultimate strain | LN | ‰ | −3.50 | −3.50 | 0.10 *** |

X_{15} | ε_{ct,max} ^{a,b,c} | Maximum tensile strain | LN | ‰ | 0.15 | 0.15 | 0.10 *** |

X_{16} | k_{1}^{b} | Tension stiffening factor (fct) | LN | 0.6 | 0.6 | 0.10 *** | |

X_{17} | k_{2}^{b} | Tension stiffening factor (εct,max) | LN | 5.0 | 5.0 | 0.10 *** | |

X_{18} | f_{y} ^{a,b,c} | Reinforcing steel yield strength | LN | MPa | 500 | 548 | 40.0 |

X_{19} | k ^{b} | Ratio (ft/fy)k for ductility class B | D | ‰ | 1.08 | 1.08 | - |

X_{20} | ε_{u} ^{b} | Strain at max. tensile stress | D | ‰ | 50 | 50 | - |

X_{21} | L ^{a,b,c} | Length | D | M | 1.92 | - | - |

X_{22} | θ_{R} ^{b} | Resistance model uncertainty | LN | - | 1.00 | 1.00 | 0.10 *** |

Variables for each step of the analysis at the system level—obtained from non-linear analysis in Sofistik software | |||||||

X_{23} | ε_{c,c} ^{b} | Concrete compressive strain | D | ‰ | Software-based | ||

X_{24} | ε_{c,t} ^{b} | Concrete tension strain | D | ‰ | Software-based | ||

X_{25} | N ^{b} | Axial acting force | D | kN | Software-based | ||

X_{26} | α_{v} ^{b} | Concrete force associated coefficient | D | / | Calculated | ||

X_{27} | k_{a} ^{b} | Concrete compressive border zone associated coefficient | D | / | Calculated | ||

X_{28} | e_{2}^{b} | Second-order eccentricity | D | Mm | Software-based |

**Table 4.**Input parameters for the probabilistic analyses of the EN 1992-1 Non-Linear Finite Element provisions of the slender column made of C100/115 (details regarding C45/55 are provided in Table 3).

X | Variable | Dist. | Unit | X_{k} | µ | σ | |
---|---|---|---|---|---|---|---|

C100/115 | |||||||

X_{1} | E_{ci} | Initial tangent concrete modulus of elasticity | LN | GPa | 48.9 | 48.9 | 6.23 |

X_{2} | E_{s} | Reinforcing steel modulus of elasticity | D. | GPa | 200 | 200 | - |

X_{11} | f_{c} | Concrete compressive strength | LN | MPa | 100.0 | 108.0 | 4.99 |

X_{12} | f_{ct} | Concrete tensile strength | LN | MPa | 3.7 | 5.2 | 1.08 |

X_{13} | G_{F} | Concrete fracture energy | LN | MPa | 119 | 170 | 35.0 |

X_{18} | f_{y} | Reinforcing steel yield strength | LN | MPa | 500 | 548 | 40.0 |

**Table 5.**Correlation between basic variables for EN 1992-1 Non-Linear Finite Element provisions of the slender columns made of C45/55 and of C100/115 [32].

C45/55 and C100/115 | |||||||
---|---|---|---|---|---|---|---|

E_{ci} | E_{s} | f_{c} | f_{ct} | G_{F} | f_{y} | ||

X_{1} | E_{ci} | 1 | 0 | 0.7 | 0.6 | 0.8 | 0 |

X_{2} | E_{s} | 1 | 0 | 0 | 0 | 0 | |

X_{11} | f_{c} | 1 | 0.9 | 0.7 | 0 | ||

X_{12} | f_{ct} | 1 | 0.5 | 0 | |||

X_{13} | G_{F} | 1 | 0 | ||||

X_{18} | f_{y} | 1 |

Safety Level | $\mathit{\beta}$ = 3.8 | $\mathit{\beta}$ = 4.2 | |||||||
---|---|---|---|---|---|---|---|---|---|

γ_{RN} | N_{d} (kN) | γ_{RM} | M_{d}(kNm) | γ_{RN} | N_{d} (kN) | γ_{RM} | M_{d}(kNm) | ||

Stiffness Method | 1.28 | 170.27 | 1.28 | 21.64 | 1.32 | 165.88 | 1.32 | 21.08 | |

NLFEM γ _{Rd} = 1.1 | BOKU | 1.37 | 246.33 | 1.32 | 17.87 | 1.41 | 238.36 | 1.36 | 17.35 |

STUBA | 1.41 | 228.90 | 1.24 | 16.11 | 1.46 | 220.81 | 1.27 | 15.75 | |

U-MINHO | 1.41 | 232.15 | 1.32 | 20.96 | 1.46 | 223.93 | 1.36 | 20.35 | |

UNIZG-FCE | 1.32 | 226.80 | 1.24 | 22.70 | 1.36 | 220.23 | 1.27 | 22.18 | |

NLFEM γ _{Rd} = 1.06 | BOKU | 1.37 | 255.67 | 1.32 | 18.54 | 1.32 | 247.30 | 1.32 | 18.00 |

STUBA | 1.41 | 237.49 | 1.24 | 16.71 | 1.41 | 229.10 | 1.36 | 16.33 | |

U-MINHO | 1.41 | 240.91 | 1.32 | 21.75 | 1.46 | 232.34 | 1.27 | 21.12 | |

UNIZG-FCE | 1.32 | 235.38 | 1.24 | 23.55 | 1.46 | 228.51 | 1.36 | 23.02 | |

Experiments | 1.24 | 235.41 | 1.28 | 20.91 | 1.23 | 230.07 | 1.32 | 20.37 |

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**MDPI and ACS Style**

Strauss, A.; Hauser, M.; Täubling, B.; Ivanković, A.M.; Skokandić, D.; Matos, J.; Galvão, N.; Benko, V.; Dobrý, J.; Wan-Wendner, R.; Ninčević, K.; Orcesi, A. Probabilistic and Semi-Probabilistic Analysis of Slender Columns Frequently Used in Structural Engineering. *Appl. Sci.* **2021**, *11*, 8009.
https://doi.org/10.3390/app11178009

**AMA Style**

Strauss A, Hauser M, Täubling B, Ivanković AM, Skokandić D, Matos J, Galvão N, Benko V, Dobrý J, Wan-Wendner R, Ninčević K, Orcesi A. Probabilistic and Semi-Probabilistic Analysis of Slender Columns Frequently Used in Structural Engineering. *Applied Sciences*. 2021; 11(17):8009.
https://doi.org/10.3390/app11178009

**Chicago/Turabian Style**

Strauss, Alfred, Michael Hauser, Benjamin Täubling, Ana Mandić Ivanković, Dominik Skokandić, José Matos, Neryvaldo Galvão, Vladimir Benko, Jakub Dobrý, Roman Wan-Wendner, Krešimir Ninčević, and André Orcesi. 2021. "Probabilistic and Semi-Probabilistic Analysis of Slender Columns Frequently Used in Structural Engineering" *Applied Sciences* 11, no. 17: 8009.
https://doi.org/10.3390/app11178009