Semi and Fully-probabilistic Nonlinear Analyses of Post-tensioned Concrete Bridge Made of KT-24 Girders

SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 23 |NUMBER: 1 |2023 |JUNE Semi and Fully-probabilistic Nonlinear Analyses of Post-tensioned Concrete Bridge Made of KT-24 Girders Bohumil ŠPLÍCHAL1, David LEHKÝ1, Jiří DOLEŽEL2 1Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic 2Moravia Consult Olomouc, a.s. Legionářská 1085/8, Olomouc, Czech Republic splic[email protected], lehky[email protected], [email protected] DOI: 10.35181/tces-2023-0005 Abstract. This paper deals with the assessment of the design resistance of an existing railway bridge made of KT-24 precast post-tensioned concrete girders. The load-bearing capacity of the structure is determined using probabilistic nonlinear analysis by the finite element method. Load-bearing capacity is determined for the ultimate and serviceability limit states. A fully probabilistic approach is compared to selected recommended semi-probabilistic methods, which can greatly reduce the number of nonlinear calculations needed to estimate the design value of resistance. Two stochastic models are compared, reflecting the level of knowledge of actual material properties from the diagnostic survey. The results are compared and discussed with respect to accuracy and required computational time, which is a critical issue when performing a global nonlinear analysis of a complex structure. Keywords Non-linear analysis, finite element method, semi-probabilistic methods, post-tensioned concrete bridge, KT-24 girders. 1. Introduction Crucial aspects of service life and reliability assessment of ageing bridges are the use of advanced material models, accurate consideration of uncertainties in the input data, and the performance of structural analyses using a global non-linear approach. The safety formats and rules normally used in codes are tailored to classical assessment procedures based on beam models, linear analysis, and local cross-section checks. Non-linear analysis, by its nature, is always a global type of assessment in which all structural parts interact. Therefore, the safety format suitable for the design of concrete structures with non-linear analysis requires a global approach (Castaldo et al. 2019 [1]). This paper compares the effect of the details of the stochastic model and the safety format method used for assessing the load-bearing capacity of an existing bridge made of KT-24 precast prestressed concrete girders. The load-bearing capacity of the structure is determined by probabilistic non-linear analysis using the finite element method. The load-bearing capacity is determined for the ultimate limit state and a couple of serviceability limit states. A fully probabilistic approach is compared with selected semi-probabilistic methods recommended by codes, which are able to significantly reduce the number of non-linear calculations required to estimate the design resistance. The methods studied include the ECoV method according to fib Model Code 2010 (fib Bulletins 65 &66 2012 [2]), the method according to EN 1992-2 (2005 [3]) and the partial safety factor method (EN 1990, 2002 [4]). These methods are also compared with a new semi-probabilistic method, which is called the Eigen ECoV method (Novák &Novák 2021 [5]), this method is based on Taylor series expansion. The results are compared and discussed in terms of accuracy and required computational time. This is a critical point when performing a global non-linear analysis of a complex structure. Two stochastic models reflecting the level of knowledge of actual material properties are used to analyze the effect of the details of the diagnostic survey on the resulting design resistance. There is a simplified ©2023 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 26 SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 23 |NUMBER: 1 |2023 |JUNE stochastic model, which represents the so-called "engineering" approach where one default input variable for each material is determined by a diagnostic survey and project documentation. The next one is a full model in which as many input variables as possible are determined according to diagnostic survey and project documentation. 2. Safety formats for global design resistance 2.1. Partial safety factor method The widely used partial safety factor method (PSF) is based on a local reduction/increase of the input parameters, which leads to structural analysis with parameters that are far from reality. This approach is not the most appropriate when combined with a global nonlinear assessment, as unrealistic input values can lead to deviations in the structural response, e.g. in failure mode. However, it is a very common method that can be used in the absence of a more suitable solution. When using the method PSF, the design resistance is calculated using the design values of the input parameters in the non-linear analysis: Rd=r(fcd, fyd, ...)(1) where r(·)represents the non-linear analysis model and the design values are calculated as fid =fik/γiM ,fik are characteristic values and γiM are partial safety factors of the materials, assumed to be 1.5 and 1.15 for concrete and steel respectively in the case of ultimate limit state and 1.0 for both materials for serviceability limit states. 2.2. EN 1992-2 method Compared to the PSF method, the EN 1992-2 method (and the other methods listed below) is based on the global safety concept. According to Eurocode 2 (EN 1992-2 2005 [3]) a design resistance Rdis calculated as: Rd=r(˜ fcm,˜ fym, ...)/γR(2) where material parameters in non-linear response function rare considered by their estimated mean values, i.e. ˜ fym = 1.1fyk (3) for mean yield strength and ˜ fcm = 1.1γs γc fck = 0.843fc(4) for mean compressive strength of concrete. Partial safety factors of steel and concrete are γs= 1.15 and γc= 1.5, respectively. The global safety factor of resistance should be considered as γR= 1.27 including model uncertainties. This value corresponds to safety index β= 3.8. The resistance function rin Equation 2 is calculated using non-linear analysis assuming above mentioned values of material properties. 2.3. ECoV Method This method mentioned in fib Model Code 2010 (fib Bulletins 65 & 66 2012 [2]) is based on idea that the structural resistance is the random variable, and its coefficient of variation VRcan be estimated from its mean Rmand characteristic values Rk, see e.g. Červenka (2013[6]). Let’s consider that structural resistance is lognormally distributed, then: VR=1 1.645ln(Rm Rk )(5) where Rm=r(fcm, fym, ...), Rk=r(fck,, fyk, ...)(6) are the mean and characteristic values of resistance which are obtained by performing two separate nonlinear analyses using mean and characteristic values of input material parameters, respectively. Global safety factor γRof resistance is then estimated as: γR=exp(αRβVR)·γRd (7) where αRis the sensitivity (weight) factor for resistance which can be considered as αR= 0.8,βis the reliability index whose value depends on the analysed limit state, see the application section, and γRd = 1.06 is the safety factor related to model and geometrical uncertainties. The design value of resistance is calculated as: Rd=Rm γk (8) The method is general and reliability level βand distribution type can be changed if required. ©2023 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 27 SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 23 |NUMBER: 1 |2023 |JUNE 2.4. Eigen ECoV method The Eigen ECoV method (Novák & Novák 2021 [5]) is derived from a Taylor series expansion with the assumption of fully correlated input random variables similarly to ECoV method according to fib Model Code 2010. The Eigen ECoV method requires 3 simulations to be performed and it is based on the idea of projecting the input random vector onto a 1D eigen distribution Θwith variance equal to the first eigenvalue of the input covariance matrix σ2 Θ=Pσ2 Xi =λ1and the mean value is simply obtained as: µΘ=v u u t N X i=1 (µ2 Xi)(9) The most effective formula for estimation of VRoffering a balance between efficiency and accuracy is in following form: VR=3Rm−4RΘδ 2+RΘδ δΘ·√λ1 Rm ,(10) where simulation RΘδ=r(XΘδ)with coordinates of input realization XΘδ= (X1δ, . . . , XNδ)and RΘδ 2= r(XΘδ 2)with coordinates of input realization XΘδ 2= (X1δ 2, . . . , XNδ 2).Rmis the mean value of resistance obtained according to Equation 6. 2.5. Probabilistic method The most advanced technique is the simulation-based probabilistic approach, which naturally reflects the uncertainties in the input data and follows the probabilistic nature of the analysis of random systems. Input parameters are considered as random variables with their prescribed correlation structure. Random realizations of input parameters are generated by a suitable sampling method, such as the Latin hypercube sampling method, which is extremely efficient for estimating statistical moments of response using a small number of samples (McKay et al. 1979 [7], Novák et al. 2014 [8]). With the set of random realizations of input parameters, a non-linear calculation is repeatedly performed, and a random global resistance of the structure is obtained. The design value of the resistance is defined such as the probability of having a more unfavourable value as follows: P(R≤Rd) = Φ(−αRβ)(11) where Φis the cumulative distribution function of the standardised Normal distribution. Probabilistic studies indicate that the random distribution of the resistance of reinforced concrete structures can be described by a two-parameter lognormal distribution with the lower bound at the origin. Then the design value of resistance is calculated as: Rd=Rm·e−αRβVR(12) where Rmand VRare the mean value and coefficient of variation, respectively, obtained from a stochastic simulation of the nonlinear model, both of which include the model uncertainties of resistance ΘR. 3. Post-tensioned concrete bridge 3.1. Basic information and computational model The post-tensioned concrete railway bridge is made of four 24 m long KT-24 girders, which are made of concrete C35/45. The mild reinforcement is made of V 10 400 (approximately Bst 420) steel and the 17 main prestressing tendons are made of 23 patented wires with a diameter of 7 mm. Figure 1 shows a schematic view of the bridge transverse section with the outer girder being modelled is marked in red. Figure 2 depicts a diagram of the reinforcement and tendons in the midspan of the girder and above the supports. Fig. 1: A schema view of the bridge transverse section. (a) Mid-span (b) Above the support Fig. 2: A schematic view of the reinforcement and tendons of the KT-24 girder in the mid-span of the girder (a) and above the supports (b). The finite element method computational model was created in ATENA 3D software (Cervenka et al. 2012 [9]). The symmetrical half of the girder was modelled. ©2023 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 28 SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 23 |NUMBER: 1 |2023 |JUNE 3D Nonlinear Cementitious 2 material model was used for concrete. Tendons and mild reinforcement were modelled using bilinear stress-strain law with hardening. The girder was loaded with self-weight, secondary dead loads, and a partial continuous load corresponding to the LM71 load model to induce the maximum bending moment (Fig. 3). Figure 4 shows the computational model including the mild reinforcement and its diameters. Figure 5 shows prestressing tendons - the mounting tendons are marked in green, and the main tendons are in red. Fig. 3: LM71 load model on the symmetrical half of the girder. Fig. 4: Computational model in ATENA 3D software – mild reinforcement. Fig. 5: Computational model in ATENA 3D software – prestressing tendons. 3.2. Stochastic models In this paper, there are used two types of stochastic models reflect the level of knowledge of actual material properties. The first is a simplified stochastic model that represents a typical situation in engineering practice, i.e., only basic diagnostics have been performed on the structure and no additional tests or supporting data/documentation are available. The model contains only one dominant input random variable for each material – compressive strength of concrete fc, the ultimate strength of prestressing tendons fup, ultimate strength of reinforcement fus, and prestressing force Pact (Tab. 1). These basic material parameters and their statistics were obtained from a diagnostic survey and JCSS Probabilistic Model Code (Joint Committee on Structural Safety 2002 [10]). Values of the remaining material parameters were derived according to fib Model Code 2010 (fib Bulletins 65 & 66 2012 [2]): fctm = 0.3·(fck)2/3(13) Ec= 22 ·(fcm ·0.1)0.3(14) Gf= 73 ·(fcm)0.18 (15) fy= 0.73 ·(fu)(16) Tab. 1: Simplified stochastic model Concrete Variable XkXmVRPDF fc[MP a]34.36 46.00 0.17 Log. 2 par. Reinforcing Variable XkXmVRPDF fus[MP a]550 611 0.05 Log. 2 par. Prestressing cable Variable XkXmVRPDF fup[MP a]1400 1456 0.025 Normal Prestressing force Variable XkXmVRPDF Pact[MP a]518.95 604 0.09 Log. 2 par. The second stochastic model is a full model, which uses as many input variables as possible, which are determined according to detailed diagnostic survey and project documentation. In this case, input variables are compressive strength of concrete fc, modulus of elasticity Ec, the ultimate strength of prestressing tendons fup, the ultimate strength of reinforcement fus and prestressing force Pact, using mean value and coefficient of variation. The mean values of the remaining variables are derived according to fib Model Code 2010 (fib Bulletins 65 & 66 2012 [2]), where every variable has its coefficient of variation. All input random variables with characteristic value, mean value, coefficient of variation and used probability distribution function are summarized in Table 2 and used correlation matrices in Tables 3 and 4 below. Note that the difference in the use of both stochastic models is not only in the number of parameters for which statistical data and probability distributions are defined but also in the way in which the individual ©2023 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 29 SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 23 |NUMBER: 1 |2023 |JUNE realizations of the variables for probabilistic and semiprobabilistic calculations are obtained. In the simplified model, only the realizations of the basic variables are generated from the probability distribution and the realizations of the other variables are then calculated from them using Equations 13–16. The consequence of this simplification is the introduction of a full stochastic correlation between the related variables. In contrast, for the full stochastic model, the realizations of all variables are generated separately from their stochastic models and the required statistical correlation according to Tables 3 and 4 is introduced by simulated annealing [11]. Tab. 2: Full stochastic model Concrete Variable XkXmVRPDF Ec[MP a]26.90 34.77 0.15 Log. 2 par. fc[MP a]34.36 46.00 0.17 Log. 2 par. ft[MP a]1.87 3.17 0.30 Log. 2 par. Gf[N/m]102.95 145.42 0.20 Log. 2 par. Reinforcing Variable XkXmVRPDF Es[GPa]200 200 - - fys[MP a]400 444 0.05 Log. 2 par. fus[MP a]550 611 0.05 Log. 2 par. Prestressing cable Variable XkXmVRPDF Ep[GPa]190 190 - - fyp[MP a]1232 1281 0.025 Normal fup[MP a]1400 1456 0.025 Normal Prestressing force Variable XkXmVRPDF Pact[MP a]518.95 604 0.09 Log. 2 par. Tab. 3: Correlation matrix of parameters of concrete EcftfcGf Ec1 0.5 0.8 0.5 ft0.5 1 0.7 0.8 fc0.8 0.7 1 0.6 Gf0.5 0.8 0.6 1 Tab. 4: Correlation matrix of parameters of reinforcement fyfu fy1 0.9 fu0.9 1 3.3. Limit states In accordance with the standard, the girder was assessed for several limit states. The one ultimate limit state and four serviceability limit states are presented here in the paper. 1) Ultimate limit state (ULS) This limit state represents the maximum load-bearing capacity of the structure, in this case, the maximum bending moment in the middle of the girder span, which is accompanied by a rapid increase in deformation, the development of major bending cracks and the progressive collapse of the entire structure, see Figure 6. In the case of ULS, a reliability index β= 3.8and safety factor related to model and geometrical uncertainties γRd = 1.06 was used according to EN 1990. Fig. 6: Girder collapse when reaching ULS and bending crack pattern. 2) Limit state of decompression (LSD) This limit state consists of verifying that the posttensioned concrete bridge is fully in a compression state to prevent cracking and the opening of transverse joints. Figure 7 shows the stress distribution when the decompression limit state is reached. For LSD, a reliability index β= 0 and safety factor related to model and geometrical uncertainties γRd = 1.0was used. Fig. 7: Stress distribution at LSD. 3) Limitation of compression stress in concrete (LCSC) The high value of compressive stress in concrete (>0.6fc) could lead to the appearance of longitu- ©2023 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 30 SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 23 |NUMBER: 1 |2023 |JUNE dinal cracks, spreading of micro-cracks in concrete and higher values of creep (mainly non-linear). Figure 8 shows the stress distribution when 60% of the compressive strength is reached in the compression zone of the girder. In the case of LCSC, a reliability index β= 1.5and safety factor related to model and geometrical uncertainties γRd = 1.02 was used. Fig. 8: Stress distribution at LCSC. 4) Limitation of tensile stress in prestressing tendons (LTSPT) The high value of tensile stress in prestressing tendons (>0.75fp) could lead to unacceptable strain. Figure 9 shows the stress distribution when 75% of the tension strength is reached in the prestressing tendons at the bottom of the girder. In the case of LTSPT, a reliability index β= 1.5and safety factor related to model and geometrical uncertainties γRd = 1.02 was used. Fig. 9: Stress distribution at LTSPT. 5) Limitation of deflection (LD) The high deflection of the girder (> l/600), which is more than 38 mm, could limit the serviceability of the structure. Figure 10 shows the vertical deflection distribution with five times scaled deformed model and line of the undeformed model when 38 mm of deflection is reached in the middle of the girder. In the case of LD, a reliability index β= 1.5and a safety factor related to model and geometrical uncertainties γRd = 1.02 was used. Fig. 10: Vertical deflection at LD. Figure 11 shows a typical diagram of mid-span deflection vs. bending moment along with an indication of reaching the above-mentioned limit states. Fig. 11: Typical loading diagram and limits of analyzed limit states. 4. Results The design value of the moment resistance was determined for all five limit states and their corresponding reliability levels using a fully probabilistic approach (FP), ECoV method, Eigen ECoV method, Partial safety factor method (PSF) and method according to EN 1992-2. In the case of the FP, 32 random realizations of the input variables were generated according to the stochastic models introduced in Tables 1 and 2. The obtained results are summarized and compared in terms of accuracy, consistency, and time consumption. 4.1. Ultimate limit state (ULS) The results of the design values of moment resistance, their corresponding quantiles, and the number of calculations (samples) required for each method are given in Table 5 for the simplified stochastic model and in Table 6 for the full stochastic model and plotted in Figure 12 for both models, where the simplified stochastic model ©2023 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 31 SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 23 |NUMBER: 1 |2023 |JUNE is in the red and full model in blue, shown distribution function a is two-dimensional lognormal probabilistic distribution function. The results show that the highest resistance was obtained by the FP approach. The semi-probabilistic methods, which are the ECoV and the Eigen ECoV along with the PSF method, provide almost similar results. The most conservative method is the EN 1992-2 method. The comparison between stochastic models shows, that only the Eigen ECoV method has different resistance, because of sensitivity of realization between characteristic and mean values. Tab. 5: Design moment resistance values for ULS at simplified model. Method Samples Resistance Quantile [kNm] [%] FP 32 16586 0.156 ECoV 2 14768 1.4 ×10−3 PSF 1 14320 1.6 ×10−4 EN 1992-2 1 12976 4.4 ×10−8 Eigen ECoV 3 14473 3.5 ×10−4 Tab. 6: Design moment resistance values for ULS at full model. Method Samples Resistance Quantile [kNm] [%] FP 32 16713 0.13 ECoV 2 14796 6 ×10−4 PSF 1 14600 1.8 ×10−5 EN 1992-2 1 13134 5 ×10−7 Eigen ECoV 3 15657 9.2 ×10−3 Fig. 12: Comparison of design moment resistance values for ULS. 4.2. Limit state of decompression (LSD) The results for LSD are listed in Table 7 for the simplified stochastic model, in Table 8 for the full stochastic model, and depicted in Figure 13 for both models (simplified is red, full is blue). In this limit state, all methods give almost the same results. This is due to the value of the reliability index β= 0, recommended for serviceability limit states with reversible effects. Tab. 7: Design moment resistance values for LSD at simplified model. Method Samples Resistance Quantile [kNm] [%] FP 32 8722 51.72 ECoV 2 8783 54.92 PSF 1 8781 54.81 Eigen ECoV 3 8783 54.92 Tab. 8: Design moment resistance values for LSD at full model. Method Samples Resistance Quantile [kNm] [%] FP 32 8686 51.77 ECoV 2 8783 56.70 PSF 1 8781 56.60 Eigen ECoV 3 8783 56.70 Fig. 13: Comparison of design moment resistance values for LSD. 4.3. Limitation of compression stress in concrete (LCSC) The results for LSCS are listed in Table 9 for the simplified stochastic model, in Table 10 for the full stochastic model and depicted in Figure 14 for both models, where the simplified model is in red and the full model is in blue. The results show that the highest resistance was again obtained by the fully probabilistic approach. The other methods give very similar values, which are approximately 8 % smaller than that given by the fully probabilistic approach. The comparison between stochastic models shows, that the fully probabilistic approach and Eigen ECoV method by the simplified model has higher resistance than the full one. The difference is circa 5 %. Other used methods have similar resistance. ©2023 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 32 SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 23 |NUMBER: 1 |2023 |JUNE Tab. 9: Design moment resistance values for LCSC at simplified model. Method Samples Resistance Quantile [kNm] [%] FP 32 12780 10.40 ECoV 2 11782 2.41 PSF 1 11810 2.54 Eigen ECoV 3 11858 2.75 Tab. 10: Design moment resistance values for LCSC at full model. Method Samples Resistance Quantile [kNm] [%] FP 32 12101 9.54 ECoV 2 11792 5.75 PSF 1 11810 5.94 Eigen ECoV 3 11434 2.89 Fig. 14: Comparison of design moment resistance values for LCSC. 4.4. Limitation of tensile stress in prestressing tendons (LTSPT) The results for LTSPT are listed in Table 11 for the simplified stochastic model, in Table 12 for the full stochastic model and depicted in Figure 15 for both models, where the simplified model is in red and the full model is in blue. The results show that the highest resistance was obtained by the Partial safety factor method for both stochastic models. This is due to the safety factor value for materials of 1.0 generally recommended for serviceability limit states. The fully probabilistic approach by full model and ECoV methods are approximately 3 % smaller than that given by the PSF method. The fully probabilistic approach by simplified stochastic model and the Eigen ECoV method for both stochastic models is 6 % smaller than the PSF method. The comparison between stochastic models shows, that the only difference is in the fully probabilistic approach, other methods have almost the same resistance. Here the reason is the lower mean resistance obtained for the simplified stochastic model. Tab. 11: Design moment resistance values for LTSPT at simplified model. Method Samples Resistance Quantile [kNm] [%] FP 32 17358 0.35 ECoV 2 17931 8.40 PSF 1 18360 33.55 Eigen ECoV 3 17039 0.03 Tab. 12: Design moment resistance values for LTSPT at full model. Method Samples Resistance Quantile [kNm] [%] FP 32 17868 1.25 ECoV 2 17850 1.15 PSF 1 18340 7.92 Eigen ECoV 3 17127 0.02 Fig. 15: Comparison of design moment resistance values for LTSPT. 4.5. Limitation of deflection (LD) The results for LD are listed in Table 13 for the simplified stochastic model, in Table 14 for the full stochastic model and depicted in Figure 16 for both models, where the simplified model is in red and the full model is in blue. The results show that the highest resistance was obtained by the Partial safety factor method for full stochastic model. The fully probabilistic approach together with other methods are approximately 8 % smaller than that given by the PSF method. The lowest resistance was obtained by Eigen ECoV by full model. The comparison between stochastic models shows, that the only difference is in the PSF and the Eigen ECoV method, other methods have almost the same resistance. ©2023 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 33 SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 23 |NUMBER: 1 |2023 |JUNE Tab. 13: Design moment resistance values for LD at simplified model. Method Samples Resistance Quantile [kNm] [%] FP 32 12029 11.50 ECoV 2 11900 8.89 PSF 1 12050 11.97 Eigen ECoV 3 11674 5.37 Tab. 14: Design moment resistance values for LD at full model. Method Samples Resistance Quantile [kNm] [%] FP 32 11837 3.84 ECoV 2 11641 2.08 PSF 1 12870 33.55 Eigen ECoV 3 10940 0.12 Fig. 16: Comparison of design moment resistance values for LD. 4.6. Discussion on number of simulations Each of the used methods requires a different number of non-linear FEM analyses and thus the required computational time. Figure 17 shows, for a simplified case of two dominant random variables of concrete compressive strength fcand tendon ultimate stress fup, how many simulations each method uses and where specific realizations are in a space of these variables. As can be seen from the Figure 17, the PSF method considers a combination of values significantly far from the area with the expected occurrence of these quantities. The EN 1992-2 method uses a combination where the steel is considered to be almost a mean value and the concrete is less than the characteristic value. The ECoV method takes into account the mean values and the characteristic values of both variables, the Eigen ECoV method additionally takes into account the intermediate values. The fully probabilistic approach performs 32 simulations, which are scattered around the mean values with respect to the given probability densities. Fig. 17: Graphical comparison of the necessary number of nonlinear FEM analyses. 5. Conclusion Non-linear fully probabilistic finite element analysis is currently the most accurate approach for determining the design resistance of structures. However, dozens of simulations are required for this approach, making it time-consuming. For this reason, simplified reliability methods that require only a few simulations are often used in engineering practice. The disadvantage of these methods is the inconsistency of the design values obtained for different structures, different limit states and failure modes, and the required reliability values. One of the main objectives of this paper was to compare the available semi-probabilistic methods together with a fully probabilistic approach in the analysis of different limit states on a post-tensioned concrete bridge. The FP method was confirmed to give the least conservative results but at the cost of increased computational time. The results for the simplified stochastic model show that the ECoV method, the Eigen ECoV method and the PSF method lead to similar values of design resistance for the ULS, LSD, LCSC and LD limit states, these are approximately 8% lower than those obtained by the FP approach for ULS and LCSC, for LSD and LD there are no differences between methods. The results for the full model look almost the same, only Eigen ECoV for the ULS, FP for LCSC and PSF and Eigen ECoV for LD have slightly different resis- ©2023 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 34