Journal of Sustainable Architecture and Civil Engineering 2018/1/22 76 *Corresponding author: aturphd@gmail.com Reliability Approaches to Modeling of the Nonlinear Pseudo-static Response of RC-structural Systems in Accidental Design Situations Received 2018/02/18 Accepted after revision 2018/06/06 Journal of Sustainable Architecture and Civil Engineering Vol. 1 / No. 22 / 2018 pp. 76-87 DOI 10.5755/j01.sace.22.1.20194 © Kaunas University of Technology Reliability Approaches to Modeling of the Nonlinear Pseudo- Static Response of RC-structural Systems in Accidental Design Situations JSACE 1/22 http://dx.doi.org/10.5755/j01.sace.22.1.20194 Andrei Tur*, Viktar Tur Brest State Technical University, Faculty of Civil Engineering, Moskovskaya 267, 224017, Brest, Republic of Belarus Introduction The nonlinear structural analysis is considered as a basic design procedure, which is used for checking of the structural robustness in accidental design situation. It is explained by following reason: a nonlinear structural analysis based on realistic constitutive relations for basic variables (average values) makes possible a simulation of a real structural behavior. It should be pointed that, implementation of the nonlinear structural analysis in design of concrete structures requires an alternate approach to safety verification. The paper presents a new approach to safety format for nonlinear analysis of RC structures subjected to accidental loads. Keywords: nonlinear analysis, progressive collapse, safety format, robustness, reliability. In recent years structural engineers try to use nonlinear analysis while designing a new complex structural system as well as for checking of the existing structures. Nonlinear analysis (static and dynamic) is most widely used as a main computational tool for checking of robustness of the structural systems in accidental design situations (Accidental Limit States Checking). As it was stated in Červenka (2013a), “evaluations of the nonlinear analysis are supported by rapid increase of computational power as well as new capabilities of the available tools for numerical simulations of structural performance”. The first published works dealing with nonlinear finite element analysis of concrete systems emerged in the late 1960. These studies focused on various aspects of element formations, including crack propagation and the bonding of reinforcement. In general case two basic FEM- methods are used for non-linear modeling: 1) so-called stiffness Method (Modified Stiffness Model); 2) Layered Model. The stiffness adaptation analysis is purposed to be an alternative for a full nonlinear analysis (Layered Model) for calculating load distributions, deformations, crack patterns and crack-width in reinforced concrete structures. In stiffness adaptation analysis both standards linear elastic material as well as non-linear mate- rial behavior can be defined (Hu and Schnobrich 1991). Nonlinear materials can be defined through 77 Journal of Sustainable Architecture and Civil Engineering 2018/1/22 a uni-axial stress-strain curve, both in the tensile and in the compressive state. Nonlinear stress- strain curves may also be defined for bar and grid reinforcement. In case of the Layered model approach each concrete layer is assumed to be in a state of plane stress and the actual stress distribution of the concrete section is modeled by a piecewise constant approximation. In general case, for the reinforcement concrete section, the final form of the stress resultant constitutive matrix at an integration point can be written as: Reliability Approaches to Modeling of the Nonlinear Pseudo-static response of RC-structural Systems in Accidental Design Situations Andrei Tur*, Viktar Tur Brest State Technical University, Faculty of Civil Engineering, Moskovskaya 267, 224017, Brest, Republic of Belarus *Corresponding author: aturphd@gmail.com The nonlinear structural analysis is considered as a basic design procedure, which is used for checking of the structural robustness in accidental design situation. It is explained by following reason: a nonlinear structural analysis based on realistic constitutive relations for basic variables (average values) makes possible a simulation of a real structural behavior. It should be pointed that, implementation of the nonlinear structural analysis in design of concrete structures requires an alternate approach to safety verification. The paper presents a new approach to safety format for nonlinear analysis of RC structures subjected to accidental loads. KEYWORDS: nonlinear analysis, progressive collapse, safety format, robustness, reliability. 1. Introduction. In recent years structural engineers try to use nonlinear analysis while designing a new complex structural system as well as for checking of the existing structures. Nonlinear analysis (static and dynamic) is most widely used as a main computational tool for checking of robustness of the structural systems in accidental design situations (Accidental Limit States Checking). As it was stated in Červenka (2013a), “evaluations of the nonlinear analysis are supported by rapid increase of computational power as well as new capabilities of the available tools for numerical simulations of structural performance”. The first published works dealing with nonlinear finite element analysis of concrete systems emerged in the late 1960. These studies focused on various aspects of element formations, including crack propagation and the bonding of reinforcement. In general case two basic FEM-methods are used for non-linear modeling: 1) so-called stiffness Method (Modified Stiffness Model); 2) Layered Model. The stiffness adaptation analysis is purposed to be an alternative for a full nonlinear analysis (Layered Model) for calculating load distributions, deformations, crack patterns and crack-width in reinforced concrete structures. In stiffness adaptation analysis both standards linear elastic material as well as non-linear material behavior can be defined (Hu and Schnobrich 1991). Nonlinear materials can be defined through a uni-axial stress-strain curve, both in the tensile and in the compressive state. Nonlinear stress-strain curves may also be defined for bar and grid reinforcement. In case of the Layered model approach each concrete layer is assumed to be in a state of plane stress and the actual stress distribution of the concrete section is modeled by a piecewise constant approximation. In general case, for the reinforcement concrete section, the final form of the stress resultant constitutive matrix at an integration point can be written as:               0 T T Rd T T Rd t T T Rd N V D M                           , (1) where:  D  is the stiffness matrix that can be established by assembling the contributions of all the concrete layers, all steel layers and transverse shear stiffness. As it was shown above, nonlinear analysis take into account the nonlinear deformation properties of RC- sections, based on physical constitutive relations (“ σ ε ” for material properties) and makes possible a simulation of a real structural behavior. If reflects an integral response, where all local sections interact and therefore it requires an adequate approach for safety verification (note, that in partial safety factor (PSF) method (EN 1990: 2006) we assume a failure probabilities of separate materials, but do not evaluate the failure probability on the structural level). It should be underlined, that nonlinear analysis offers a verification of global resistance and requires a safety format for global resistance (Červenka 2013b). In accordance with Červenka (2013b), the term global resistance (global safety) is used for “assessment of structural response on higher structural level than a cross-section”. The term global resistance is introduced in Červenka (2013b) in order to distinguish the newly introduced check of safety on global level, as compared to local safety check in the partial safety factor method (PSF-method) in accordance with EN 1990 (2006). (1) where: [ ]D – is the stiffness matrix that can be established by assembling the contribu- tions of all the concrete layers, all steel layers and transverse shear stiffness. As it was shown above, nonlinear analysis take into account the nonlinear deformation proper- ties of RC-sections, based on physical constitutive relations (“ ó å− ” for material properties) and makes possible a simulation of a real structural behavior. If reflects an integral response, where all local sections interact and therefore it requires an adequate approach for safety verification (note, that in partial safety factor (PSF) method (EN 1990: 2006) we assume a failure probabilities of separate materials, but do not evaluate the failure probability on the structural level). It should be underlined, that nonlinear analysis offers a verification of global resistance and requires a safety format for global resistance (Červenka 2013b). In accordance with Červenka (2013b), the term global resistance (global safety) is used for “assessment of structural response on higher structural level than a cross-section”. The term global resistance is introduced in Červenka (2013b) in order to distinguish the newly introduced check of safety on global level, as compared to local safety check in the partial safety factor method (PSF-method) in accordance with EN 1990 (2006). he historical review (from CEM MC78 to fib MC2010) of the non-linear safety format development was described in detail in Sangiorgio (2015). With the implementation of the new fib MC2010 (2010), a different perspective was placed on nonlinear analysis and safety assessment. The design condition to be used in safety format for nonlinear analysis is written in the external actions and resisting internal forces domain: 2. Safety format for nonlinear analysis in accordance with actual codes provision. The historical review (from CEM MC78 to fib MC2010) of the non-linear safety format development was described in detail in Sangiorgio (2015). With the implementation of the new fib MC2010 (2010), a different perspective was placed on nonlinear analysis and safety assessment. The design condition to be used in safety format for nonlinear analysis is written in the external actions and resisting internal forces domain: d dE R , (2) where: dE  is the design value of the action; dR  is the design value of resistance. Three different approaches are proposed to evaluate the design resistance dR (depending on various levels of implementations of probabilistic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) the global resistance method; and (3) the partial factor method (PSF-method). In the global resistance format, the resistance is considered on a global structural level. Two alternative methods are mentioned in fib MC2010 (2010) for the derivation of the design resistance dR : (1) global resistance factor method (which was adopted from EN 1992-2 (2005), slightly modified); and (2) ECOV-method, proposed by Červenka (2013b) and Sykora and Holicky (2011) (estimations of coefficient variation for resistance). In this case, the safety margin can be expressed by the global safety factor as: md R R R   , (3) where: mR  is the mean resistance. The global safety factor R cover all uncertainties and can be related to the coefficient of variations of resistance RV (according a LN- distribution (!) according EN 1992-2 (2005)) as exp( )R R RV    . A simplified formulation was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq. (3) is product of two factors R m Rd    (Sykora and Holicky 2011). The first factor m is related to material uncertainty and can be established by probabilistic analysis. The second factor Rd is related to model and geometrical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2 (2005)). As it was stated in Sangiorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, only few contributions were found in literature (Schlune et al. 2011, Allaix et al. 2013). The first contribution was presented by Schlune et al. (2011). Design resistance dR is then derived by division of the obtained mean resistance mR by global resistance factor R : ( , , )ym cm nom d R R f f a R   , (4) where: exp( )R R R m V      again based on the assumption of a lognormal distributed resistance. Model uncertainties are explicitly taken into accounts thought the use of the bias factor θm , which is defined as the mean ratio of experimental to predicted resistance (in accordance with Schlune et al. (2011)) its value varies between 0.7 and 1.2 for failure in compression, bending and shear). The coefficient variation of structural resistance RV is written as follows: 2 2 2R g m fV V V V   , (5) where: , ,g m fV V V  are the coefficients of variations of the geometrical, model and material uncertainties respectively, estimated in accordance with Schlune et al. (2011). According with the second contributions, proposed by (Allaix and Mancini 2007, Allaix et al. 2013) the design resistance dR is derived by divisions of obtained resistance factor R and the model uncertainty factor Rd : ( , , )ym cm nom d R Rd R f f a R    , (6) In this case, the global resistance factor R is derived from coefficient of variations of the structural resistance RV (estimated by probabilistic method or based on Červenka method ECOV): exp( )R R RV    , (7) where:  is the reliability index in accordance with EN 1990; R - is the sensitivity factor for resistance. (2) where: dE – is the design value of the action; dR – is the design value of resistance. Three different approaches are proposed to evaluate the design resistance dR (depending on var- ious levels of implementations of probabilistic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) the global resistance method; and (3) the partial factor method (PSF-method). In the global resistance format, the resistance is considered on a global structural level. Two alter- native methods are mentioned in fib MC2010 (2010) for the derivation of the design resistance dR : (1) global resistance factor method (which was adopted from EN 1992-2 (2005), slightly modified); and (2) ECOV-method, proposed by Červenka (2013b) and Sykora and Holicky (2011) (estimations of coefficient variation for resistance). In this case, the safety margin can be expressed by the global safety factor as: 2. Safety format for nonlinear analysis in accordance with actual codes provision. The historical review (from CEM MC78 to fib MC2010) of the non-linear safety format development was described in detail in Sangiorgio (2015). With the implementation of the new fib MC2010 (2010), a different perspective was placed on nonlinear analysis and safety assessment. The design condition to be used in safety format for nonlinear analysis is written in the external actions and resisting internal forces domain: d dE R , (2) where: dE  is the design value of the action; dR  is the design value of resistance. Three different approaches are proposed to evaluate the design resistance dR (depending on various levels of implementations of probabilistic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) the global resistance method; and (3) the partial factor method (PSF-method). In the global resistance format, the resistance is considered on a global structural level. Two alternative methods are mentioned in fib MC2010 (2010) for the derivation of the design resistance dR : (1) global resistance factor method (which was adopted from EN 1992-2 (2005), slightly modified); and (2) ECOV-method, proposed by Červenka (2013b) and Sykora and Holicky (2011) (estimations of coefficient variation for resistance). In this case, the safety margin can be expressed by the global safety factor as: md R R R   , (3) where: mR  is the mean resistance. The global safety factor R cover all uncertainties and can be related to the coefficient of variations of resistance RV (according a LN- distribution (!) according EN 1992-2 (2005)) as exp( )R R RV    . A simplified formulation was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq. (3) is product of two factors R m Rd    (Sykora and Holicky 2011). The first factor m is related to material uncertainty and can be established by probabilistic analysis. The second factor Rd is related to model and geometrical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2 (2005)). As it was stated in Sangiorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, only few contributions were found in literature (Schlune et al. 2011, Allaix et al. 2013). The first contribution was presented by Schlune et al. (2011). Design resistance dR is then derived by division of the obtained mean resistance mR by global resistance factor R : ( , , )ym cm nom d R R f f a R   , (4) where: exp( )R R R m V      again based on the assumption of a lognormal distributed resistance. Model uncertainties are explicitly taken into accounts thought the use of the bias factor θm , which is defined as the mean ratio of experimental to predicted resistance (in accordance with Schlune et al. (2011)) its value varies between 0.7 and 1.2 for failure in compression, bending and shear). The coefficient variation of structural resistance RV is written as follows: 2 2 2R g m fV V V V   , (5) where: , ,g m fV V V  are the coefficients of variations of the geometrical, model and material uncertainties respectively, estimated in accordance with Schlune et al. (2011). According with the second contributions, proposed by (Allaix and Mancini 2007, Allaix et al. 2013) the design resistance dR is derived by divisions of obtained resistance factor R and the model uncertainty factor Rd : ( , , )ym cm nom d R Rd R f f a R    , (6) In this case, the global resistance factor R is derived from coefficient of variations of the structural resistance RV (estimated by probabilistic method or based on Červenka method ECOV): exp( )R R RV    , (7) where:  is the reliability index in accordance with EN 1990; R - is the sensitivity factor for resistance. (3) where: mR – is the mean resistance. The global safety factor Rγ cover all uncertainties and can be related to the coefficient of variations Safety format for nonlinear analysis in accordance with actual codes provision Journal of Sustainable Architecture and Civil Engineering 2018/1/22 78 of resistance RV (according a LN- distribution (!) according EN 1992-2 (2005)) as exp( )R R RVγ = α β . A simplified formulation was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq. (3) is product of two factors R m Rdγ = γ ⋅γ (Sykora and Holicky 2011). The first factor mγ is related to material uncertainty and can be established by probabilistic analysis. The second factor Rdγ is related to model and geometrical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2(2005)). As it was stated in Sangiorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, only few contributions were found in literature (Schlune et al. 2011, Allaix et al. 2013). The first contribution was presented by Schlune et al. (2011). Design resistance dR is then derived by division of the obtained mean resistance mR by global resistance factor Rγ : 2. Safety format for nonlinear analysis in accordance with actual codes provision. The historical review (from CEM MC78 to fib MC2010) of the non-linear safety format development was described in detail in Sangiorgio (2015). With the implementation of the new fib MC2010 (2010), a different perspective was placed on nonlinear analysis and safety assessment. The design condition to be used in safety format for nonlinear analysis is written in the external actions and resisting internal forces domain: d dE R , (2) where: dE  is the design value of the action; dR  is the design value of resistance. Three different approaches are proposed to evaluate the design resistance dR (depending on various levels of implementations of probabilistic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) the global resistance method; and (3) the partial factor method (PSF-method). In the global resistance format, the resistance is considered on a global structural level. Two alternative methods are mentioned in fib MC2010 (2010) for the derivation of the design resistance dR : (1) global resistance factor method (which was adopted from EN 1992-2 (2005), slightly modified); and (2) ECOV-method, proposed by Červenka (2013b) and Sykora and Holicky (2011) (estimations of coefficient variation for resistance). In this case, the safety margin can be expressed by the global safety factor as: md R R R   , (3) where: mR  is the mean resistance. The global safety factor R cover all uncertainties and can be related to the coefficient of variations of resistance RV (according a LN- distribution (!) according EN 1992-2 (2005)) as exp( )R R RV    . A simplified formulation was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq. (3) is product of two factors R m Rd    (Sykora and Holicky 2011). The first factor m is related to material uncertainty and can be established by probabilistic analysis. The second factor Rd is related to model and geometrical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2 (2005)). As it was stated in Sangiorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, only few contributions were found in literature (Schlune et al. 2011, Allaix et al. 2013). The first contribution was presented by Schlune et al. (2011). Design resistance dR is then derived by division of the obtained mean resistance mR by global resistance factor R : ( , , )ym cm nom d R R f f a R   , (4) where: exp( )R R R m V      again based on the assumption of a lognormal distributed resistance. Model uncertainties are explicitly taken into accounts thought the use of the bias factor θm , which is defined as the mean ratio of experimental to predicted resistance (in accordance with Schlune et al. (2011)) its value varies between 0.7 and 1.2 for failure in compression, bending and shear). The coefficient variation of structural resistance RV is written as follows: 2 2 2R g m fV V V V   , (5) where: , ,g m fV V V  are the coefficients of variations of the geometrical, model and material uncertainties respectively, estimated in accordance with Schlune et al. (2011). According with the second contributions, proposed by (Allaix and Mancini 2007, Allaix et al. 2013) the design resistance dR is derived by divisions of obtained resistance factor R and the model uncertainty factor Rd : ( , , )ym cm nom d R Rd R f f a R    , (6) In this case, the global resistance factor R is derived from coefficient of variations of the structural resistance RV (estimated by probabilistic method or based on Červenka method ECOV): exp( )R R RV    , (7) where:  is the reliability index in accordance with EN 1990; R - is the sensitivity factor for resistance. (4) where: exp( )R R R m Vα β γ = θ – again based on the assumption of a lognormal distributed resis- tance. Model uncertainties are explicitly taken into accounts thought the use of the bias factor è m , which is defined as the mean ratio of experimental to predicted resistance (in accordance with Schlune et al. (2011)) its value varies between 0.7 and 1.2 for failure in compression, bending and shear). The coefficient variation of structural resistance RV is written as follows: 2. Safety format for nonlinear analysis in accordance with actual codes provision. The historical review (from CEM MC78 to fib MC2010) of the non-linear safety format development was described in detail in Sangiorgio (2015). With the implementation of the new fib MC2010 (2010), a different perspective was placed on nonlinear analysis and safety assessment. The design condition to be used in safety format for nonlinear analysis is written in the external actions and resisting internal forces domain: d dE R , (2) where: dE  is the design value of the action; dR  is the design value of resistance. Three different approaches are proposed to evaluate the design resistance dR (depending on various levels of implementations of probabilistic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) the global resistance method; and (3) the partial factor method (PSF-method). In the global resistance format, the resistance is considered on a global structural level. Two alternative methods are mentioned in fib MC2010 (2010) for the derivation of the design resistance dR : (1) global resistance factor method (which was adopted from EN 1992-2 (2005), slightly modified); and (2) ECOV-method, proposed by Červenka (2013b) and Sykora and Holicky (2011) (estimations of coefficient variation for resistance). In this case, the safety margin can be expressed by the global safety factor as: md R R R   , (3) where: mR  is the mean resistance. The global safety factor R cover all uncertainties and can be related to the coefficient of variations of resistance RV (according a LN- distribution (!) according EN 1992-2 (2005)) as exp( )R R RV    . A simplified formulation was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq. (3) is product of two factors R m Rd    (Sykora and Holicky 2011). The first factor m is related to material uncertainty and can be established by probabilistic analysis. The second factor Rd is related to model and geometrical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2 (2005)). As it was stated in Sangiorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, only few contributions were found in literature (Schlune et al. 2011, Allaix et al. 2013). The first contribution was presented by Schlune et al. (2011). Design resistance dR is then derived by division of the obtained mean resistance mR by global resistance factor R : ( , , )ym cm nom d R R f f a R   , (4) where: exp( )R R R m V      again based on the assumption of a lognormal distributed resistance. Model uncertainties are explicitly taken into accounts thought the use of the bias factor θm , which is defined as the mean ratio of experimental to predicted resistance (in accordance with Schlune et al. (2011)) its value varies between 0.7 and 1.2 for failure in compression, bending and shear). The coefficient variation of structural resistance RV is written as follows: 2 2 2R g m fV V V V   , (5) where: , ,g m fV V V  are the coefficients of variations of the geometrical, model and material uncertainties respectively, estimated in accordance with Schlune et al. (2011). According with the second contributions, proposed by (Allaix and Mancini 2007, Allaix et al. 2013) the design resistance dR is derived by divisions of obtained resistance factor R and the model uncertainty factor Rd : ( , , )ym cm nom d R Rd R f f a R    , (6) In this case, the global resistance factor R is derived from coefficient of variations of the structural resistance RV (estimated by probabilistic method or based on Červenka method ECOV): exp( )R R RV    , (7) where:  is the reliability index in accordance with EN 1990; R - is the sensitivity factor for resistance. (5) where: , ,g m fV V V – are the coefficients of vari- ations of the geometrical, model and material uncertainties respectively, estimated in accor- dance with Schlune et al. (2011). According with the second contributions, proposed by (Allaix and Mancini 2007, Allaix et al. 2013) the design resistance dR is derived by divisions of obtained resistance factor Rγ and the model uncertainty factor Rdγ : 2. Safety format for nonlinear analysis in accordance with actual codes provision. The historical review (from CEM MC78 to fib MC2010) of the non-linear safety format development was described in detail in Sangiorgio (2015). With the implementation of the new fib MC2010 (2010), a different perspective was placed on nonlinear analysis and safety assessment. The design condition to be used in safety format for nonlinear analysis is written in the external actions and resisting internal forces domain: d dE R , (2) where: dE  is the design value of the action; dR  is the design value of resistance. Three different approaches are proposed to evaluate the design resistance dR (depending on various levels of implementations of probabilistic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) the global resistance method; and (3) the partial factor method (PSF-method). In the global resistance format, the resistance is considered on a global structural level. Two alternative methods are mentioned in fib MC2010 (2010) for the derivation of the design resistance dR : (1) global resistance factor method (which was adopted from EN 1992-2 (2005), slightly modified); and (2) ECOV-method, proposed by Červenka (2013b) and Sykora and Holicky (2011) (estimations of coefficient variation for resistance). In this case, the safety margin can be expressed by the global safety factor as: md R R R   , (3) where: mR  is the mean resistance. The global safety factor R cover all uncertainties and can be related to the coefficient of variations of resistance RV (according a LN- distribution (!) according EN 1992-2 (2005)) as exp( )R R RV    . A simplified formulation was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq. (3) is product of two factors R m Rd    (Sykora and Holicky 2011). The first factor m is related to material uncertainty and can be established by probabilistic analysis. The second factor Rd is related to model and geometrical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2 (2005)). As it was stated in Sangiorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, only few contributions were found in literature (Schlune et al. 2011, Allaix et al. 2013). The first contribution was presented by Schlune et al. (2011). Design resistance dR is then derived by division of the obtained mean resistance mR by global resistance factor R : ( , , )ym cm nom d R R f f a R   , (4) where: exp( )R R R m V      again based on the assumption of a lognormal distributed resistance. Model uncertainties are explicitly taken into accounts thought the use of the bias factor θm , which is defined as the mean ratio of experimental to predicted resistance (in accordance with Schlune et al. (2011)) its value varies between 0.7 and 1.2 for failure in compression, bending and shear). The coefficient variation of structural resistance RV is written as follows: 2 2 2R g m fV V V V   , (5) where: , ,g m fV V V  are the coefficients of variations of the geometrical, model and material uncertainties respectively, estimated in accordance with Schlune et al. (2011). According with the second contributions, proposed by (Allaix and Mancini 2007, Allaix et al. 2013) the design resistance dR is derived by divisions of obtained resistance factor R and the model uncertainty factor Rd : ( , , )ym cm nom d R Rd R f f a R    , (6) In this case, the global resistance factor R is derived from coefficient of variations of the structural resistance RV (estimated by probabilistic method or based on Červenka method ECOV): exp( )R R RV    , (7) where:  is the reliability index in accordance with EN 1990; R - is the sensitivity factor for resistance. (6) In this case, the global resistance factor Rγ is derived from coefficient of variations of the structur- al resistance RV (estimated by probabilistic method or based on Červenka method ECOV): 2. Safety format for nonlinear analysis in accordance with actual codes provision. The historical review (from CEM MC78 to fib MC2010) of the non-linear safety format development was described in detail in Sangiorgio (2015). With the implementation of the new fib MC2010 (2010), a different perspective was placed on nonlinear analysis and safety assessment. The design condition to be used in safety format for nonlinear analysis is written in the external actions and resisting internal forces domain: d dE R , (2) where: dE  is the design value of the action; dR  is the design value of resistance. Three different approaches are proposed to evaluate the design resistance dR (depending on various levels of implementations of probabilistic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) the global resistance method; and (3) the partial factor method (PSF-method). In the global resistance format, the resistance is considered on a global structural level. Two alternative methods are mentioned in fib MC2010 (2010) for the derivation of the design resistance dR : (1) global resistance factor method (which was adopted from EN 1992-2 (2005), slightly modified); and (2) ECOV-method, proposed by Červenka (2013b) and Sykora and Holicky (2011) (estimations of coefficient variation for resistance). In this case, the safety margin can be expressed by the global safety factor as: md R R R   , (3) where: mR  is the mean resistance. The global safety factor R cover all uncertainties and can be related to the coefficient of variations of resistance RV (according a LN- distribution (!) according EN 1992-2 (2005)) as exp( )R R RV    . A simplified formulation was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq. (3) is product of two factors R m Rd    (Sykora and Holicky 2011). The first factor m is related to material uncertainty and can be established by probabilistic analysis. The second factor Rd is related to model and geometrical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2 (2005)). As it was stated in Sangiorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, only few contributions were found in literature (Schlune et al. 2011, Allaix et al. 2013). The first contribution was presented by Schlune et al. (2011). Design resistance dR is then derived by division of the obtained mean resistance mR by global resistance factor R : ( , , )ym cm nom d R R f f a R   , (4) where: exp( )R R R m V      again based on the assumption of a lognormal distributed resistance. Model uncertainties are explicitly taken into accounts thought the use of the bias factor θm , which is defined as the mean ratio of experimental to predicted resistance (in accordance with Schlune et al. (2011)) its value varies between 0.7 and 1.2 for failure in compression, bending and shear). The coefficient variation of structural resistance RV is written as follows: 2 2 2R g m fV V V V   , (5) where: , ,g m fV V V  are the coefficients of variations of the geometrical, model and material uncertainties respectively, estimated in accordance with Schlune et al. (2011). According with the second contributions, proposed by (Allaix and Mancini 2007, Allaix et al. 2013) the design resistance dR is derived by divisions of obtained resistance factor R and the model uncertainty factor Rd : ( , , )ym cm nom d R Rd R f f a R    , (6) In this case, the global resistance factor R is derived from coefficient of variations of the structural resistance RV (estimated by probabilistic method or based on Červenka method ECOV): exp( )R R RV    , (7) where:  is the reliability index in accordance with EN 1990; R - is the sensitivity factor for resistance. (7) where: β is the reliability index in accordance with EN 1990; Rα – is the sensitivity factor for resistance. The model uncertainty factor Rdγ takes into account the difference between the real behavior of the structure and the results obtained based on a numerical model. The model uncertainty factor Rdγ can be derived using the following expression from Schlune et al. (2011): The model uncertainty factor Rd takes into account the difference between the real behavior of the structure and the results obtained based on a numerical model. The model uncertainty factor Rd can be derived using the following expression from Schlune et al. (2011): exp( )Rd R RV    , (8) where: 0, 4R R    is the sensitivity factor for resistance model uncertainty ( R <1 in order to account for separate safety assessment of resistance); RV  is the coefficient of variations of the resistance model uncertainly. The value of this coefficient of variations can be obtained based on experimental results according to EN 1990 (2006). For checking of the RC-structural system in accidental design situation, two main issues must be solved: (1) to calculate the pseudo-static response of the modified structural system under accidental loads; (2) to determined target value of the reliability index for accidental design (required level of reliability). 3. Pseudo-static response of the structural system with a removed vertical load bearing elements. As was stated in Ellingwood (2002) prevent and mitigation of progressive collapse can be achieved using two different methods: (1) TF-method (indirect Tie-Force method); (2) AP – method (direct Alternate Load Path method). The indirect (TF - method) consists of improving the structural integrity of building by providing redundancy of load path and ductile detailing. Currently, the EN 1991-1-7, allows the use of indirect method and some guidance is contained in the EN 1992-1-1. In this case criteria are devised to check the local resistance to withstand a specific postulated accidental load. The direct method, referred to as “Alternate Load Path” (AP - method), is most widely used in the practical design and based on criteria for evaluating the capability of a damaged structure to bridge over or around the damaged volume of area without progressive collapse developing from the local damage. The AP-method consists in considering internal force (effect of the actions) redistributions throughout the structure following the loss of a vertical support element (UFC 4-023-03: 2010). As was shown in UFC 4-023-03 (2010), an AP-method analysis may be performed using of the following basic nonlinear procedures: Nonlinear Dynamic (NLD) and Nonlinear Static (NLS) procedures. In case of the Nonlinear Static procedure after materially-and-geometrically nonlinear model is built, the accidental load combination are magnified by a dynamic increase factor (DIF) that accounts for inertia effects and the resulting load is applied to model with removed vertical load bearing elements. If a dynamic increase factor (DIF) is known, for deformation- controlled actions, the resulting deformations are compared to the expected deformation capacities; for the force controlled action, the member strength is not modified and shell not be less than the maximum internal member forces (demands). Otherwise, calculation procedure based on the energetic approach should be used. The basic provisions of this procedure are described in detail in Tur (2012). The purpose here is to analyze the structural response of RC-structural systems subjected to a sudden column loss. The procedure, which is used for obtaining of the pseudo-static nonlinear response of the structural system, consists of the following main steps: (1) Calculate the static non-linear response “ F δ ” for the modified structural system with a removed vertical load bearing element according to certain rules (Tur 2012, Vlassis 2009) (see Fig. 1, line 1); (2) Calculate the pseudo-static response, that taking into account inertia effects, caused by suddenly applied gravity load. In general case, based on energetic consideration (see Fig. 1): , 0 ( ) u ps u uP d F      , (9) (8) where: 0, 4R Rα = α – is the sensitivity factor for resistance model uncertainty ( Rα <1 in order to account for separate safety assessment of re- sistance); RVϑ – is the coefficient of variations of the resistance model uncertainly. The value of this coefficient of variations can be obtained based on experimental results according to EN 1990 (2006). 79 Journal of Sustainable Architecture and Civil Engineering 2018/1/22 For checking of the RC-structural system in accidental design situation, two main issues must be solved: (1) to calculate the pseudo-static response of the modified structural system under accidental loads; (2) to determined target value of the reliability index for accidental design (required level of reliability). As was stated in Ellingwood (2002) prevent and mitigation of progressive collapse can be achieved using two different methods: (1) TF-method (indirect Tie-Force method); (2) AP – method (direct Al- ternate Load Path method). The indirect (TF - method) consists of improving the structural integrity of building by providing redundancy of load path and ductile detailing. Currently, the EN 1991-1-7, allows the use of indirect method and some guidance is contained in the EN 1992-1-1. In this case criteria are devised to check the local resistance to withstand a specific postulated accidental load. The direct method, referred to as “Alternate Load Path” (AP - method), is most widely used in the practical design and based on criteria for evaluating the capability of a damaged structure to bridge over or around the damaged volume of area without progressive collapse developing from the local damage. The AP-method consists in considering internal force (effect of the actions) redistributions throughout the structure following the loss of a vertical support element (UFC 4-023-03: 2010). As was shown in UFC 4-023-03 (2010), an AP-method analysis may be performed using of the following basic nonlinear procedures: Nonlinear Dynamic (NLD) and Nonlinear Static (NLS) pro- cedures. In case of the Nonlinear Static procedure after materially-and-geometrically nonlinear model is built, the accidental load combination are magnified by a dynamic increase factor (DIF) that accounts for inertia effects and the resulting load is applied to model with removed vertical load bearing elements. If a dynamic increase factor (DIF) is known, for deformation-controlled actions, the resulting deformations are compared to the expected deformation capacities; for the force controlled action, the member strength is not modified and shell not be less than the maximum internal member forces (demands). Otherwise, calculation procedure based on the energetic approach should be used. The basic provisions of this procedure are described in detail in Tur (2012). The purpose here is to analyze the structural response of RC-structural systems subjected to a sudden column loss. The procedure, which is used for obtaining of the pseudo-static nonlinear response of the struc- tural system, consists of the following main steps: (1) Calculate the static non-linear response “F–δ” for the modified structural system with a removed vertical load bearing element according to certain rules (Tur 2012, Vlassis 2009) (see Fig. 1, line 1); (2) Calculate the pseudo-static re- Pseudo- static response of the structural system with a removed vertical load bearing elements Fig. 1 To assessment of the pseudo-static response of the structural system in accordance with (Tur 2012, Vlassis 2009) Journal of Sustainable Architecture and Civil Engineering 2018/1/22 80 sponse, that taking into account inertia effects, caused by suddenly applied gravity load. In general case, based on energetic consideration (see Fig. 1): The model uncertainty factor Rd takes into account the difference between the real behavior of the structure and the results obtained based on a numerical model. The model uncertainty factor Rd can be derived using the following expression from Schlune et al. (2011): exp( )Rd R RV    , (8) where: 0, 4R R    is the sensitivity factor for resistance model uncertainty ( R <1 in order to account for separate safety assessment of resistance); RV  is the coefficient of variations of the resistance model uncertainly. The value of this coefficient of variations can be obtained based on experimental results according to EN 1990 (2006). For checking of the RC-structural system in accidental design situation, two main issues must be solved: (1) to calculate the pseudo-static response of the modified structural system under accidental loads; (2) to determined target value of the reliability index for accidental design (required level of reliability). 3. Pseudo-static response of the structural system with a removed vertical load bearing elements. As was stated in Ellingwood (2002) prevent and mitigation of progressive collapse can be achieved using two different methods: (1) TF-method (indirect Tie-Force method); (2) AP – method (direct Alternate Load Path method). The indirect (TF - method) consists of improving the structural integrity of building by providing redundancy of load path and ductile detailing. Currently, the EN 1991-1-7, allows the use of indirect method and some guidance is contained in the EN 1992-1-1. In this case criteria are devised to check the local resistance to withstand a specific postulated accidental load. The direct method, referred to as “Alternate Load Path” (AP - method), is most widely used in the practical design and based on criteria for evaluating the capability of a damaged structure to bridge over or around the damaged volume of area without progressive collapse developing from the local damage. The AP-method consists in considering internal force (effect of the actions) redistributions throughout the structure following the loss of a vertical support element (UFC 4-023-03: 2010). As was shown in UFC 4-023-03 (2010), an AP-method analysis may be performed using of the following basic nonlinear procedures: Nonlinear Dynamic (NLD) and Nonlinear Static (NLS) procedures. In case of the Nonlinear Static procedure after materially-and-geometrically nonlinear model is built, the accidental load combination are magnified by a dynamic increase factor (DIF) that accounts for inertia effects and the resulting load is applied to model with removed vertical load bearing elements. If a dynamic increase factor (DIF) is known, for deformation- controlled actions, the resulting deformations are compared to the expected deformation capacities; for the force controlled action, the member strength is not modified and shell not be less than the maximum internal member forces (demands). Otherwise, calculation procedure based on the energetic approach should be used. The basic provisions of this procedure are described in detail in Tur (2012). The purpose here is to analyze the structural response of RC-structural systems subjected to a sudden column loss. The procedure, which is used for obtaining of the pseudo-static nonlinear response of the structural system, consists of the following main steps: (1) Calculate the static non-linear response “ F δ ” for the modified structural system with a removed vertical load bearing element according to certain rules (Tur 2012, Vlassis 2009) (see Fig. 1, line 1); (2) Calculate the pseudo-static response, that taking into account inertia effects, caused by suddenly applied gravity load. In general case, based on energetic consideration (see Fig. 1): , 0 ( ) u ps u uP d F      , (9) (9) Pseudo-static response is equal: Fig. 1. To assessment of the pseudo-static response of the structural system in accordance with (Tur 2012, Vlassis 2009) Pseudo-static response is equal: , 0 1 ( ) u ps u u F P d       , (10) 4. Required level of reliability for accidental design situation. In general case, the probability of structure collapse due to postulated abnormal event can be written as: ( ) ( ) ( ) ( )i i iP F P F DH P D H P H , (11) As was shown in Ellingwood (2002), in a “specific local resistance” design strategy, the focus is on minimizing probability ( )iP F DH , that is, to minimize the likelihood of initiation of damage that may lead to progressive collapse. This strategy may be difficult or uneconomical, and may leave some significant hazards unaddressed. Accordingly, it is likely that ( )iP D H will very close to 1,0 in many practical cases, meaning that the collapse probability becomes, approximately: ( ) ( ) ( )i iP F P F DH P H , (12) It is in minimizing the conditional probability ( )iP F H , that the science and art of the structural engineer becomes paramount (Ellingwood 2002). It may be assumed that the occurrence of the abnormal event iH can be modeled as a Poisson process with yearly mean rate of occurrence i . The probability of occurrence of this abnormal event during some reference period T, is thus approximately ( )i iP H T  (for very small i ) (Ellingwood 2002). In the case of fire, gas explosion and some other accidental loads, parameter i may be related to building floor area ( i fp A   , in which fA - floor area and 1 2p p p  , where term 1p  probability of occurrence of hazard per unit area and 2 1.0p  represents effect of warning and control systems). Mean rates of occurrence for gas explosions, bomb explosions and vehicular collisions in accordance with (Ellingwood and Corotis 1991) are approximately: Gas explosions (per dwelling): 2x10-5/yr; Bomb explosions (per dwelling): 2x10-6/yr; Vehicular collisions (per dwelling): 6x10-4/yr; Full developed fire (per building): 5x10-8/yr. As it was shown in (Ellingwood 2002), to evaluate ( )iP F DH , one must postulate a mathematical model, G(X) (state model), of the structural system based on principles of mechanics and supplemented, where possible, with experimental data (!). The load and resistance variables are expressed by vector X. We must then determine the probability distribution of each variable and integrate the joint density function of X over that region of probability space where G(X) <0 to compute in accordance with EN 1990 conventional limit state probability. But, we must to remember that it is very difficult and complex way (especially for structural systems). Alternatively, FORM – analysis may be used to compute a conditional reliability index  defined as: G G     , (13) where: G and G  is mean and standard deviation of G(X). According to Ellingwood (2002), the reliability index is related to ( )iP F DH through: -1[ ( )]iP F DH   , (14) in which -1[ ( )]iP F DH is the percent-point function of the standard Normal probability distribution. With ( )i iP H T  , eq. (14) can be rewritten as: -1[ ( / )]iP F T    , (15) As was shown in Ellingwood (2002), the first-generation probability-based Limit State Design Criteria (such as, for example, EUROCODES) all are based, to varying degrees, on reliability of individual structural members and components. However, to implement reliability-based design criteria against progressive collapse in practice sense, the limit state probability (or reliability index) must be evaluated for a structural system (!). In contrast to member reliability, (10) In general case, the probability of structure collapse due to postulated abnormal event can be written as: Fig. 1. To assessment of the pseudo-static response of the structural system in accordance with (Tur 2012, Vlassis 2009) Pseudo-static response is equal: , 0 1 ( ) u ps u u F P d       , (10) 4. Required level of reliability for accidental design situation. In general case, the probability of structure collapse due to postulated abnormal event can be written as: ( ) ( ) ( ) ( )i i iP F P F DH P D H P H , (11) As was shown in Ellingwood (2002), in a “specific local resistance” design strategy, the focus is on minimizing probability ( )iP F DH , that is, to minimize the likelihood of initiation of damage that may lead to progressive collapse. This strategy may be difficult or uneconomical, and may leave some significant hazards unaddressed. Accordingly, it is likely that ( )iP D H will very close to 1,0 in many practical cases, meaning that the collapse probability becomes, approximately: ( ) ( ) ( )i iP F P F DH P H , (12) It is in minimizing the conditional probability ( )iP F H , that the science and art of the structural engineer becomes paramount (Ellingwood 2002). It may be assumed that the occurrence of the abnormal event iH can be modeled as a Poisson process with yearly mean rate of occurrence i . The probability of occurrence of this abnormal event during some reference period T, is thus approximately ( )i iP H T  (for very small i ) (Ellingwood 2002). In the case of fire, gas explosion and some other accidental loads, parameter i may be related to building floor area ( i fp A   , in which fA - floor area and 1 2p p p  , where term 1p  probability of occurrence of hazard per unit area and 2 1.0p  represents effect of warning and control systems). Mean rates of occurrence for gas explosions, bomb explosions and vehicular collisions in accordance with (Ellingwood and Corotis 1991) are approximately: Gas explosions (per dwelling): 2x10-5/yr; Bomb explosions (per dwelling): 2x10-6/yr; Vehicular collisions (per dwelling): 6x10-4/yr; Full developed fire (per building): 5x10-8/yr. As it was shown in (Ellingwood 2002), to evaluate ( )iP F DH , one must postulate a mathematical model, G(X) (state model), of the structural system based on principles of mechanics and supplemented, where possible, with experimental data (!). The load and resistance variables are expressed by vector X. We must then determine the probability distribution of each variable and integrate the joint density function of X over that region of probability space where G(X) <0 to compute in accordance with EN 1990 conventional limit state probability. But, we must to remember that it is very difficult and complex way (especially for structural systems). Alternatively, FORM – analysis may be used to compute a conditional reliability index  defined as: G G     , (13) where: G and G  is mean and standard deviation of G(X). According to Ellingwood (2002), the reliability index is related to ( )iP F DH through: -1[ ( )]iP F DH   , (14) in which -1[ ( )]iP F DH is the percent-point function of the standard Normal probability distribution. With ( )i iP H T  , eq. (14) can be rewritten as: -1[ ( / )]iP F T    , (15) As was shown in Ellingwood (2002), the first-generation probability-based Limit State Design Criteria (such as, for example, EUROCODES) all are based, to varying degrees, on reliability of individual structural members and components. However, to implement reliability-based design criteria against progressive collapse in practice sense, the limit state probability (or reliability index) must be evaluated for a structural system (!). In contrast to member reliability, (11) As was shown in Ellingwood (2002), in a “specific local resistance” design strategy, the focus is on minimizing probability ( )iP F DH , that is, to minimize the likelihood of initiation of damage that may lead to progressive collapse. This strategy may be difficult or uneconomical, and may leave some significant hazards unad- dressed. Accordingly, it is likely that ( )iP D H will very close to 1,0 in many practical cases, meaning that the collapse probability becomes, approximately: Fig. 1. To assessment of the pseudo-static response of the structural system in accordance with (Tur 2012, Vlassis 2009) Pseudo-static response is equal: , 0 1 ( ) u ps u u F P d       , (10) 4. Required level of reliability for accidental design situation. In general case, the probability of structure collapse due to postulated abnormal event can be written as: ( ) ( ) ( ) ( )i i iP F P F DH P D H P H , (11) As was shown in Ellingwood (2002), in a “specific local resistance” design strategy, the focus is on minimizing probability ( )iP F DH , that is, to minimize the likelihood of initiation of damage that may lead to progressive collapse. This strategy may be difficult or uneconomical, and may leave some significant hazards unaddressed. Accordingly, it is likely that ( )iP D H will very close to 1,0 in many practical cases, meaning that the collapse probability becomes, approximately: ( ) ( ) ( )i iP F P F DH P H , (12) It is in minimizing the conditional probability ( )iP F H , that the science and art of the structural engineer becomes paramount (Ellingwood 2002). It may be assumed that the occurrence of the abnormal event iH can be modeled as a Poisson process with yearly mean rate of occurrence i . The probability of occurrence of this abnormal event during some reference period T, is thus approximately ( )i iP H T  (for very small i ) (Ellingwood 2002). In the case of fire, gas explosion and some other accidental loads, parameter i may be related to building floor area ( i fp A   , in which fA - floor area and 1 2p p p  , where term 1p  probability of occurrence of hazard per unit area and 2 1.0p  represents effect of warning and control systems). Mean rates of occurrence for gas explosions, bomb explosions and vehicular collisions in accordance with (Ellingwood and Corotis 1991) are approximately: Gas explosions (per dwelling): 2x10-5/yr; Bomb explosions (per dwelling): 2x10-6/yr; Vehicular collisions (per dwelling): 6x10-4/yr; Full developed fire (per building): 5x10-8/yr. As it was shown in (Ellingwood 2002), to evaluate ( )iP F DH , one must postulate a mathematical model, G(X) (state model), of the structural system based on principles of mechanics and supplemented, where possible, with experimental data (!). The load and resistance variables are expressed by vector X. We must then determine the probability distribution of each variable and integrate the joint density function of X over that region of probability space where G(X) <0 to compute in accordance with EN 1990 conventional limit state probability. But, we must to remember that it is very difficult and complex way (especially for structural systems). Alternatively, FORM – analysis may be used to compute a conditional reliability index  defined as: G G     , (13) where: G and G  is mean and standard deviation of G(X). According to Ellingwood (2002), the reliability index is related to ( )iP F DH through: -1[ ( )]iP F DH   , (14) in which -1[ ( )]iP F DH is the percent-point function of the standard Normal probability distribution. With ( )i iP H T  , eq. (14) can be rewritten as: -1[ ( / )]iP F T    , (15) As was shown in Ellingwood (2002), the first-generation probability-based Limit State Design Criteria (such as, for example, EUROCODES) all are based, to varying degrees, on reliability of individual structural members and components. However, to implement reliability-based design criteria against progressive collapse in practice sense, the limit state probability (or reliability index) must be evaluated for a structural system (!). In contrast to member reliability, (12) It is in minimizing the conditional probability ( )iP F H , that the science and art of the structural engineer becomes paramount (Ellingwood 2002). It may be assumed that the occurrence of the abnormal event iH can be modeled as a Poisson process with yearly mean rate of occurrence iλ . The probability of occurrence of this abnormal event during some reference period T, is thus approximately ( )i iP H T= λ (for very small iλ ) (Ellingwood 2002). In the case of fire, gas explosion and some other accidental loads, parameter iλ may be related to building floor area ( i fp Aλ = ⋅ , in which fA - floor area and 1 2p p p= ⋅ , where term 1p – probability of occurrence of hazard per unit area and 2 1.0p < represents effect of warning and control systems). Mean rates of occurrence for gas explosions, bomb explosions and vehicular collisions in accor- dance with (Ellingwood and Corotis 1991) are approximately: _ Gas explosions (per dwelling): 2x10-5/yr; _ Bomb explosions (per dwelling): 2x10-6/yr; _ Vehicular collisions (per dwelling): 6x10-4/yr; _ Full developed fire (per building): 5x10-8/yr. As it was shown in (Ellingwood 2002), to evaluate ( )iP F DH , one must postulate a mathe- matical model, G(X) (state model), of the structural system based on principles of mechanics and supplemented, where possible, with experimental data (!). The load and resistance variables are expressed by vector X. We must then determine the probability distribution of each variable and integrate the joint density function of X over that region of probability space where G(X) <0 Required level of reliability for accidental design situation 81 Journal of Sustainable Architecture and Civil Engineering 2018/1/22 to compute in accordance with EN 1990 conventional limit state probability. But, we must to remember that it is very difficult and complex way (especially for structural systems). Alternatively, FORM – analysis may be used to compute a conditional reliability index β defined as: Fig. 1. To assessment of the pseudo-static response of the structural system in accordance with (Tur 2012, Vlassis 2009) Pseudo-static response is equal: , 0 1 ( ) u ps u u F P d       , (10) 4. Required level of reliability for accidental design situation. In general case, the probability of structure collapse due to postulated abnormal event can be written as: ( ) ( ) ( ) ( )i i iP F P F DH P D H P H , (11) As was shown in Ellingwood (2002), in a “specific local resistance” design strategy, the focus is on minimizing probability ( )iP F DH , that is, to minimize the likelihood of initiation of damage that may lead to progressive collapse. This strategy may be difficult or uneconomical, and may leave some significant hazards unaddressed. Accordingly, it is likely that ( )iP D H will very close to 1,0 in many practical cases, meaning that the collapse probability becomes, approximately: ( ) ( ) ( )i iP F P F DH P H , (12) It is in minimizing the conditional probability ( )iP F H , that the science and art of the structural engineer becomes paramount (Ellingwood 2002). It may be assumed that the occurrence of the abnormal event iH can be modeled as a Poisson process with yearly mean rate of occurrence i . The probability of occurrence of this abnormal event during some reference period T, is thus approximately ( )i iP H T  (for very small i ) (Ellingwood 2002). In the case of fire, gas explosion and some other accidental loads, parameter i may be related to building floor area ( i fp A   , in which fA - floor area and 1 2p p p  , where term 1p  probability of occurrence of hazard per unit area and 2 1.0p  represents effect of warning and control systems). Mean rates of occurrence for gas explosions, bomb explosions and vehicular collisions in accordance with (Ellingwood and Corotis 1991) are approximately: Gas explosions (per dwelling): 2x10-5/yr; Bomb explosions (per dwelling): 2x10-6/yr; Vehicular collisions (per dwelling): 6x10-4/yr; Full developed fire (per building): 5x10-8/yr. As it was shown in (Ellingwood 2002), to evaluate ( )iP F DH , one must postulate a mathematical model, G(X) (state model), of the structural system based on principles of mechanics and supplemented, where possible, with experimental data (!). The load and resistance variables are expressed by vector X. We must then determine the probability distribution of each variable and integrate the joint density function of X over that region of probability space where G(X) <0 to compute in accordance with EN 1990 conventional limit state probability. But, we must to remember that it is very difficult and complex way (especially for structural systems). Alternatively, FORM – analysis may be used to compute a conditional reliability index  defined as: G G     , (13) where: G and G  is mean and standard deviation of G(X). According to Ellingwood (2002), the reliability index is related to ( )iP F DH through: -1[ ( )]iP F DH   , (14) in which -1[ ( )]iP F DH is the percent-point function of the standard Normal probability distribution. With ( )i iP H T  , eq. (14) can be rewritten as: -1[ ( / )]iP F T    , (15) As was shown in Ellingwood (2002), the first-generation probability-based Limit State Design Criteria (such as, for example, EUROCODES) all are based, to varying degrees, on reliability of individual structural members and components. However, to implement reliability-based design criteria against progressive collapse in practice sense, the limit state probability (or reliability index) must be evaluated for a structural system (!). In contrast to member reliability, (13) where: Gµ and Gσ – is mean and standard deviation of G(X). According to Ellingwood (2002), the reliability index is related to ( )iP F DH through: Fig. 1. To assessment of the pseudo-static response of the structural system in accordance with (Tur 2012, Vlassis 2009) Pseudo-static response is equal: , 0 1 ( ) u ps u u F P d       , (10) 4. Required level of reliability for accidental design situation. In general case, the probability of structure collapse due to postulated abnormal event can be written as: ( ) ( ) ( ) ( )i i iP F P F DH P D H P H , (11) As was shown in Ellingwood (2002), in a “specific local resistance” design strategy, the focus is on minimizing probability ( )iP F DH , that is, to minimize the likelihood of initiation of damage that may lead to progressive collapse. This strategy may be difficult or uneconomical, and may leave some significant hazards unaddressed. Accordingly, it is likely that ( )iP D H will very close to 1,0 in many practical cases, meaning that the collapse probability becomes, approximately: ( ) ( ) ( )i iP F P F DH P H , (12) It is in minimizing the conditional probability ( )iP F H , that the science and art of the structural engineer becomes paramount (Ellingwood 2002). It may be assumed that the occurrence of the abnormal event iH can be modeled as a Poisson process with yearly mean rate of occurrence i . The probability of occurrence of this abnormal event during some reference period T, is thus approximately ( )i iP H T  (for very small i ) (Ellingwood 2002). In the case of fire, gas explosion and some other accidental loads, parameter i may be related to building floor area ( i fp A   , in which fA - floor area and 1 2p p p  , where term 1p  probability of occurrence of hazard per unit area and 2 1.0p  represents effect of warning and control systems). Mean rates of occurrence for gas explosions, bomb explosions and vehicular collisions in accordance with (Ellingwood and Corotis 1991) are approximately: Gas explosions (per dwelling): 2x10-5/yr; Bomb explosions (per dwelling): 2x10-6/yr; Vehicular collisions (per dwelling): 6x10-4/yr; Full developed fire (per building): 5x10-8/yr. As it was shown in (Ellingwood 2002), to evaluate ( )iP F DH , one must postulate a mathematical model, G(X) (state model), of the structural system based on principles of mechanics and supplemented, where possible, with experimental data (!). The load and resistance variables are expressed by vector X. We must then determine the probability distribution of each variable and integrate the joint density function of X over that region of probability space where G(X) <0 to compute in accordance with EN 1990 conventional limit state probability. But, we must to remember that it is very difficult and complex way (especially for structural systems). Alternatively, FORM – analysis may be used to compute a conditional reliability index  defined as: G G     , (13) where: G and G  is mean and standard deviation of G(X). According to Ellingwood (2002), the reliability index is related to ( )iP F DH through: -1[ ( )]iP F DH   , (14) in which -1[ ( )]iP F DH is the percent-point function of the standard Normal probability distribution. With ( )i iP H T  , eq. (14) can be rewritten as: -1[ ( / )]iP F T    , (15) As was shown in Ellingwood (2002), the first-generation probability-based Limit State Design Criteria (such as, for example, EUROCODES) all are based, to varying degrees, on reliability of individual structural members and components. However, to implement reliability-based design criteria against progressive collapse in practice sense, the limit state probability (or reliability index) must be evaluated for a structural system (!). In contrast to member reliability, (14) in which -1[ ( )]iP F DHΦ is the percent-point function of the standard Normal probability distribution. With ( )i iP H T= λ , eq. (14) can be rewritten as: Fig. 1. To assessment of the pseudo-static response of the structural system in accordance with (Tur 2012, Vlassis 2009) Pseudo-static response is equal: , 0 1 ( ) u ps u u F P d       , (10) 4. Required level of reliability for accidental design situation. In general case, the probability of structure collapse due to postulated abnormal event can be written as: ( ) ( ) ( ) ( )i i iP F P F DH P D H P H , (11) As was shown in Ellingwood (2002), in a “specific local resistance” design strategy, the focus is on minimizing probability ( )iP F DH , that is, to minimize the likelihood of initiation of damage that may lead to progressive collapse. This strategy may be difficult or uneconomical, and may leave some significant hazards unaddressed. Accordingly, it is likely that ( )iP D H will very close to 1,0 in many practical cases, meaning that the collapse probability becomes, approximately: ( ) ( ) ( )i iP F P F DH P H , (12) It is in minimizing the conditional probability ( )iP F H , that the science and art of the structural engineer becomes paramount (Ellingwood 2002). It may be assumed that the occurrence of the abnormal event iH can be modeled as a Poisson process with yearly mean rate of occurrence i . The probability of occurrence of this abnormal event during some reference period T, is thus approximately ( )i iP H T  (for very small i ) (Ellingwood 2002). In the case of fire, gas explosion and some other accidental loads, parameter i may be related to building floor area ( i fp A   , in which fA - floor area and 1 2p p p  , where term 1p  probability of occurrence of hazard per unit area and 2 1.0p  represents effect of warning and control systems). Mean rates of occurrence for gas explosions, bomb explosions and vehicular collisions in accordance with (Ellingwood and Corotis 1991) are approximately: Gas explosions (per dwelling): 2x10-5/yr; Bomb explosions (per dwelling): 2x10-6/yr; Vehicular collisions (per dwelling): 6x10-4/yr; Full developed fire (per building): 5x10-8/yr. As it was shown in (Ellingwood 2002), to evaluate ( )iP F DH , one must postulate a mathematical model, G(X) (state model), of the structural system based on principles of mechanics and supplemented, where possible, with experimental data (!). The load and resistance variables are expressed by vector X. We must then determine the probability distribution of each variable and integrate the joint density function of X over that region of probability space where G(X) <0 to compute in accordance with EN 1990 conventional limit state probability. But, we must to remember that it is very difficult and complex way (especially for structural systems). Alternatively, FORM – analysis may be used to compute a conditional reliability index  defined as: G G     , (13) where: G and G  is mean and standard deviation of G(X). According to Ellingwood (2002), the reliability index is related to ( )iP F DH through: -1[ ( )]iP F DH   , (14) in which -1[ ( )]iP F DH is the percent-point function of the standard Normal probability distribution. With ( )i iP H T  , eq. (14) can be rewritten as: -1[ ( / )]iP F T    , (15) As was shown in Ellingwood (2002), the first-generation probability-based Limit State Design Criteria (such as, for example, EUROCODES) all are based, to varying degrees, on reliability of individual structural members and components. However, to implement reliability-based design criteria against progressive collapse in practice sense, the limit state probability (or reliability index) must be evaluated for a structural system (!). In contrast to member reliability, (15) As was shown in Ellingwood (2002), the first-generation probability-based Limit State Design Cri- teria (such as, for example, EUROCODES) all are based, to varying degrees, on reliability of indi- vidual structural members and components. However, to implement reliability-based design criteria against progressive collapse in practice sense, the limit state probability (or reliability index) must be evaluated for a structural system (!). In contrast to member reliability, this evaluation is difficult (!) even at the present state of art and with computational resources available (Ellingwood 2002, Tur 2012). Assuming that an analysis of a damaged structure can be performed, an acceptable value of β upon which to base design for conditional limit states is suggested by eq. (15). As shown by Ellingwood (2002), the probability of structural system failure is an order of magni- tude less, depending on the redundancy in the system and the degree continuity between mem- bers. For example, if 610i −λ = to 10-5, than the conditional failure probability for the structural system should be on the order of 10-2…10-1, and the target value of reliability index tagβ should be the order of 1,5. Load and resistance criteria can be developed to be consistent with the reliability. At the first stage of analysis the value of the global resistance factor Rγ was defined in accordance with (Sykora and Holicky 2011) from eq. (7). As it was shown above, the ECOV-method is based on idea that the random distribution of resistance, which is again described by the coefficient varia- tion VR, can be estimated from mean Rm and characteristic Rk values of resistance (pseudo-static response of the structural system). In this case, coefficient variations of resistance VR can be ob- tained from following equation: this evaluation is difficult (!) even at the present state of art and with computational resources available (Ellingwood 2002, Tur 2012). Assuming that an analysis of a damaged structure can be performed, an acceptable value of  upon which to base design for conditional limit states is suggested by eq. (15). As shown by Ellingwood (2002), the probability of structural system failure is an order of magnitude less, depending on the redundancy in the system and the degree continuity between members. For example, if 610i   to 10-5, than the conditional failure probability for the structural system should be on the order of 10-2…10-1, and the target value of reliability index tag should be the order of 1,5. Load and resistance criteria can be developed to be consistent with the reliability. 5. Assessment of the global resistance and global safety factors for pseudo-static response. At the first stage of analysis the value of the global resistance factor R was defined in accordance with (Sykora and Holicky 2011) from eq. (7). As it was shown above, the ECOV-method is based on idea that the random distribution of resistance, which is again described by the coefficient variation VR, can be estimated from mean Rm and characteristic Rk values of resistance (pseudo-static response of the structural system). In this case, coefficient variations of resistance VR can be obtained from following equation: 1 ln( ) 1, 64 m R k R V R  , (16) where: ,m kR R  are the mean and characteristic values of resistance (pseudo-static response, as was shown in section 3), obtained by two separate non-linear analysis using mean and characteristic values of input material parameters respectively. The results of the nonlinear analysis of the statically undetermined an encastre RC-beam and values of the coefficient variations VR and global resistance coefficient R obtained by calculations are presented in Table 1. Table 1. The results of estimation the coefficient R based on ECOV Element l ' l r r [%] Resistance, kN/m 1 ln( ) 1, 64 m R k R V R  exp(1, 2 )R RV  Rm Rk RC-beam (an encastre) 0,48 1,05 119,5 109,4 0,054 1,07 Notes: Materials properties: concrete class C25/30, fcm=33 MPa, steel B500, fym=1,1fyk=550 MPa; section 300x350 mm; 1, 5tag  for accidental design situation. The result, presented in Table 1 was obtained with FEM-computer program most widely used in practical design and declared about possibilities for nonlinear analysis of reinforced concrete structures. As it was declared in software manual, FE-program is capable of a “realistic simulation of RC-structure” behavior in the entire loading range with ductile as well as brittle failure modes (Sykora and Holicky 2011, Schlune et al. 2011). As was shown in Allaix and Mancini (2007) the result of investigation depends on assumption and criteria underlying the model used in the non-linear analysis. It should be noted that the different FEM-programs (software), which applied for nonlinear structural analysis, will have own different level of FEM-model uncertainties in addition to local cross-section resistance model, material and geometry uncertainties. Clearly, the approach is meaningful if structural model covers all relevant failure mechanisms. So, effects of model uncertainties should be treated separately (!). At the second stage of analysis the coefficient of variations RV of the computer model uncertainties was assessed based on theoretical background described in Annex D (EN 1990: 2006). From these features, it is suggested to be derived from the comparison of the experimental tests data and numerical calculations results, but though probabilistic consideration. The set of the test results obtained in experimental investigations of the different types of statically indeterminate structures demonstrates different failure mechanism (see Tables 2, 3) was collected from some references and used for assessment of the coefficient variations RV and model uncertainly factor Rd . The model uncertainty factor Rd takes into account difference between the real behavior of structure and the results of a numerical modeling suitable for specific structure. The real properties of the material and specimens geometry characteristics obtained by testing were used as an input data for nonlinear analysis. The main characteristics of the analyzed test specimens are presented in Tables 2, 3. As it can be seen from the Table 4, the estimated values of coefficient of variations VRd for model uncertainties are much higher than recommended in codes (for example, in fib MC2010, values in range 1,05…1,1). (16) where: ,m kR R – are the mean and character- istic values of resistance (pseudo-static re- sponse, as was shown in section 3), obtained by two separate non-linear analysis using mean and characteristic values of input ma- terial parameters respectively. The results of the nonlinear analysis of the statically undetermined an encastre RC-beam and values of the coefficient variations VR and global resistance coefficient Rγ obtained by calculations are presented in Table 1. The result, presented in Table 1 was obtained with FEM-computer program most widely used in practical design and declared about possibilities for nonlinear analysis of reinforced concrete Assessment of the global resistance and global safety factors for pseudo- static response Journal of Sustainable Architecture and Civil Engineering 2018/1/22 82 structures. As it was declared in software manual, FE-program is capable of a “realistic simulation of RC-structure” behavior in the entire loading range with ductile as well as brittle failure modes (Sykora and Holicky 2011, Schlune et al. 2011). As was shown in Allaix and Mancini (2007) the result of investigation depends on assumption and criteria underlying the model used in the non-linear analysis. It should be noted that the different FEM-programs (software), which applied for nonlinear structural analysis, will have own different level of FEM-model uncertainties in addition to local cross-section resistance model, material and geometry uncertainties. Clearly, the approach is meaningful if structural model covers all relevant failure mechanisms. So, effects of model uncertainties should be treated separately (!). Table 1 The results of estimation the coefficient Rγ based on ECOV Element l ' l r r [%] Resistance, kN/m 1 ln( ) 1, 64 m R k R V R = exp(1, 2 )R RVγ = Rm Rk RC-beam (an encastre) 0,48 1,05 119,5 109,4 0,054 1,07 Notes: Materials properties: concrete class C25/30, fcm=33 MPa, steel B500, fym=1,1fyk=550 MPa; section 300x350 mm; 1, 5tagβ = for accidental design situation. Table 2 Loading arrangement for experimental specimens Loading scheme Beam, slabs Series Reference The same results and conclusions were obtained by Schlune et al. (2011). Schlune concluded that model uncertainties of nonlinear analysis are much higher than in standard design based on engineering formulas and are strongly dependent on modes of failure and adopted failure criteria. Reported in Schlune et al. (2011) coefficient of variation due to model uncertainty for bending failure in range 5…30%, for shear 15…64%. Schlune concluded that due to the lack of data, the choice of model uncertainty often depends on engineering judgment and can be subjective. Table 2. Loading arrangement for experimental specimens Loading scheme Beam, slabs Series Reference B1, B2, B3 Monnier (1970) B4, B6, B7, B8, B9 Saleh and Barem (2013), Ashour and Habeeb (2008), Maghsoudi and Bengar (2009), Mahmoud and Afefy (2012), Dalfré and Barros (2011) B5 Saleh and Barem (2013) B10, B11, B13 Farhangvesali et al. (2013), Parmar et al. (2015) B12 Qian and Li (2012) B14, B15 Rashidian et al. (2016) Slab S7, S11, S12, S15, S16, S17, S27, S28, S33 Cardenas and Sozen (1968) Cylindrical Shell SH11, SH31 Duddeck et. al. (1978) Note, that coefficient of variations Vm due to material uncertainty (variability) has not a fixed value. In the case of concrete, the mean value of the concrete compressive strength for different classes according to EN 1992-2 (2005) is calculated as: fcm=fck+8 MPa (where 8 1, 64 cMPa   , which standard deviation 4, 88c MPa  ). For fixed value of standard deviation (as a basic characteristic of the production quality control) 4, 5c MPa  , coefficient of variation Vm,c of concrete compressive strength will be in range from 8,6 % (C50/60) to 21 % (C16/20) and coefficient of variation for materials Vm will be in range from Vm=10,48 % to 21,84 % (with fixed value of coefficient of variations Vs=6 % for steel). B1, B2, B3 Monnier (1970) The same results and conclusions were obtained by Schlune et al. (2011). Schlune concluded that model uncertainties of nonlinear analysis are much higher than in standard design based on engineering formulas and are strongly dependent on modes of failure and adopted failure criteria. Reported in Schlune et al. (2011) coefficient of variation due to model uncertainty for bending failure in range 5…30%, for shear 15…64%. Schlune concluded that due to the lack of data, the choice of model uncertainty often depends on engineering judgment and can be subjective. Table 2. Loading arrangement for experimental specimens Loading scheme Beam, slabs Series Reference B1, B2, B3 Monnier (1970) B4, B6, B7, B8, B9 Saleh and Barem (2013), Ashour and Habeeb (2008), Maghsoudi and Bengar (2009), Mahmoud and Afefy (2012), Dalfré and Barros (2011) B5 Saleh and Barem (2013) B10, B11, B13 Farhangvesali et al. (2013), Parmar et al. (2015) B12 Qian and Li (2012) B14, B15 Rashidian et al. (2016) Slab S7, S11, S12, S15, S16, S17, S27, S28, S33 Cardenas and Sozen (1968) Cylindrical Shell SH11, SH31 Duddeck et. al. (1978) Note, that coefficient of variations Vm due to material uncertainty (variability) has not a fixed value. In the case of concrete, the mean value of the concrete compressive strength for different classes according to EN 1992-2 (2005) is calculated as: fcm=fck+8 MPa (where 8 1, 64 cMPa   , which standard deviation 4, 88c MPa  ). For fixed value of standard deviation (as a basic characteristic of the production quality control) 4, 5c MPa  , coefficient of variation Vm,c of concrete compressive strength will be in range from 8,6 % (C50/60) to 21 % (C16/20) and coefficient of variation for materials Vm will be in range from Vm=10,48 % to 21,84 % (with fixed value of coefficient of variations Vs=6 % for steel). B4, B6, B7, B8, B9 Saleh and Barem (2013), Ashour and Habeeb (2008), Maghsoudi and Bengar (2009), Mahmoud and Afefy (2012), Dalfré and Barros (2011) The same results and conclusions were obtained by Schlune et al. (2011). Schlune concluded that model uncertainties of nonlinear analysis are much higher than in standard design based on engineering formulas and are strongly dependent on modes of failure and adopted failure criteria. Reported in Schlune et al. (2011) coefficient of variation due to model uncertainty for bending failure in range 5…30%, for shear 15…64%. Schlune concluded that due to the lack of data, the choice of model uncertainty often depends on engineering judgment and can be subjective. Table 2. Loading arrangement for experimental specimens Loading scheme Beam, slabs Series Reference B1, B2, B3 Monnier (1970) B4, B6, B7, B8, B9 Saleh and Barem (2013), Ashour and Habeeb (2008), Maghsoudi and Bengar (2009), Mahmoud and Afefy (2012), Dalfré and Barros (2011) B5 Saleh and Barem (2013) B10, B11, B13 Farhangvesali et al. (2013), Parmar et al. (2015) B12 Qian and Li (2012) B14, B15 Rashidian et al. (2016) Slab S7, S11, S12, S15, S16, S17, S27, S28, S33 Cardenas and Sozen (1968) Cylindrical Shell SH11, SH31 Duddeck et. al. (1978) Note, that coefficient of variations Vm due to material uncertainty (variability) has not a fixed value. In the case of concrete, the mean value of the concrete compressive strength for different classes according to EN 1992-2 (2005) is calculated as: fcm=fck+8 MPa (where 8 1, 64 cMPa   , which standard deviation 4, 88c MPa  ). For fixed value of standard deviation (as a basic characteristic of the production quality control) 4, 5c MPa  , coefficient of variation Vm,c of concrete compressive strength will be in range from 8,6 % (C50/60) to 21 % (C16/20) and coefficient of variation for materials Vm will be in range from Vm=10,48 % to 21,84 % (with fixed value of coefficient of variations Vs=6 % for steel). B5 Saleh and Barem (2013) The same results and conclusions were obtained by Schlune et al. (2011). Schlune concluded that model uncertainties of nonlinear analysis are much higher than in standard design based on engineering formulas and are strongly dependent on modes of failure and adopted failure criteria. Reported in Schlune et al. (2011) coefficient of variation due to model uncertainty for bending failure in range 5…30%, for shear 15…64%. Schlune concluded that due to the lack of data, the choice of model uncertainty often depends on engineering judgment and can be subjective. Table 2. Loading arrangement for experimental specimens Loading scheme Beam, slabs Series Reference B1, B2, B3 Monnier (1970) B4, B6, B7, B8, B9 Saleh and Barem (2013), Ashour and Habeeb (2008), Maghsoudi and Bengar (2009), Mahmoud and Afefy (2012), Dalfré and Barros (2011) B5 Saleh and Barem (2013) B10, B11, B13 Farhangvesali et al. (2013), Parmar et al. (2015) B12 Qian and Li (2012) B14, B15 Rashidian et al. (2016) Slab S7, S11, S12, S15, S16, S17, S27, S28, S33 Cardenas and Sozen (1968) Cylindrical Shell SH11, SH31 Duddeck et. al. (1978) Note, that coefficient of variations Vm due to material uncertainty (variability) has not a fixed value. In the case of concrete, the mean value of the concrete compressive strength for different classes according to EN 1992-2 (2005) is calculated as: fcm=fck+8 MPa (where 8 1, 64 cMPa   , which standard deviation 4, 88c MPa  ). For fixed value of standard deviation (as a basic characteristic of the production quality control) 4, 5c MPa  , coefficient of variation Vm,c of concrete compressive strength will be in range from 8,6 % (C50/60) to 21 % (C16/20) and coefficient of variation for materials Vm will be in range from Vm=10,48 % to 21,84 % (with fixed value of coefficient of variations Vs=6 % for steel). B10, B11, B13 Farhangvesali et al. (2013), Parmar et al. (2015) The same results and conclusions were obtained by Schlune et al. (2011). Schlune concluded that model uncertainties of nonlinear analysis are much higher than in standard design based on engineering formulas and are strongly dependent on modes of failure and adopted failure criteria. Reported in Schlune et al. (2011) coefficient of variation due to model uncertainty for bending failure in range 5…30%, for shear 15…64%. Schlune concluded that due to the lack of data, the choice of model uncertainty often depends on engineering judgment and can be subjective. Table 2. Loading arrangement for experimental specimens Loading scheme Beam, slabs Series Reference B1, B2, B3 Monnier (1970) B4, B6, B7, B8, B9 Saleh and Barem (2013), Ashour and Habeeb (2008), Maghsoudi and Bengar (2009), Mahmoud and Afefy (2012), Dalfré and Barros (2011) B5 Saleh and Barem (2013) B10, B11, B13 Farhangvesali et al. (2013), Parmar et al. (2015) B12 Qian and Li (2012) B14, B15 Rashidian et al. (2016) Slab S7, S11, S12, S15, S16, S17, S27, S28, S33 Cardenas and Sozen (1968) Cylindrical Shell SH11, SH31 Duddeck et. al. (1978) Note, that coefficient of variations Vm due to material uncertainty (variability) has not a fixed value. In the case of concrete, the mean value of the concrete compressive strength for different classes according to EN 1992-2 (2005) is calculated as: fcm=fck+8 MPa (where 8 1, 64 cMPa   , which standard deviation 4, 88c MPa  ). For fixed value of standard deviation (as a basic characteristic of the production quality control) 4, 5c MPa  , coefficient of variation Vm,c of concrete compressive strength will be in range from 8,6 % (C50/60) to 21 % (C16/20) and coefficient of variation for materials Vm will be in range from Vm=10,48 % to 21,84 % (with fixed value of coefficient of variations Vs=6 % for steel). B12 Qian and Li (2012) The same results and conclusions were obtained by Schlune et al. (2011). Schlune concluded that model uncertainties of nonlinear analysis are much higher than in standard design based on engineering formulas and are strongly dependent on modes of failure and adopted failure criteria. Reported in Schlune et al. (2011) coefficient of variation due to model uncertainty for bending failure in range 5…30%, for shear 15…64%. Schlune concluded that due to the lack of data, the choice of model uncertainty often depends on engineering judgment and can be subjective. Table 2. Loading arrangement for experimental specimens Loading scheme Beam, slabs Series Reference B1, B2, B3 Monnier (1970) B4, B6, B7, B8, B9 Saleh and Barem (2013), Ashour and Habeeb (2008), Maghsoudi and Bengar (2009), Mahmoud and Afefy (2012), Dalfré and Barros (2011) B5 Saleh and Barem (2013) B10, B11, B13 Farhangvesali et al. (2013), Parmar et al. (2015) B12 Qian and Li (2012) B14, B15 Rashidian et al. (2016) Slab S7, S11, S12, S15, S16, S17, S27, S28, S33 Cardenas and Sozen (1968) Cylindrical Shell SH11, SH31 Duddeck et. al. (1978) Note, that coefficient of variations Vm due to material uncertainty (variability) has not a fixed value. In the case of concrete, the mean value of the concrete compressive strength for different classes according to EN 1992-2 (2005) is calculated as: fcm=fck+8 MPa (where 8 1, 64 cMPa   , which standard deviation 4, 88c MPa  ). For fixed value of standard deviation (as a basic characteristic of the production quality control) 4, 5c MPa  , coefficient of variation Vm,c of concrete compressive strength will be in range from 8,6 % (C50/60) to 21 % (C16/20) and coefficient of variation for materials Vm will be in range from Vm=10,48 % to 21,84 % (with fixed value of coefficient of variations Vs=6 % for steel). B14, B15 Rashidian et al. (2016) Slab S7, S11, S12, S15, S16, S17, S27, S28, S33 Cardenas and Sozen (1968) Cylindrical Shell SH11, SH31 Duddeck et. al. (1978) 83 Journal of Sustainable Architecture and Civil Engineering 2018/1/22 At the second stage of analysis the coefficient of variations RVϑ of the computer model uncertain- ties was assessed based on theoretical background described in Annex D (EN 1990: 2006). From these features, it is suggested to be derived from the comparison of the experimental tests data and numerical calculations results, but though probabilistic consideration. The set of the test results obtained in experimental investigations of the different types of stat- ically indeterminate structures demonstrates different failure mechanism (see Tables 2, 3) was Table 3 Basic parameters of the experimental specimens (input data for non-linear analysis)B ea m /S la b S er ie s Cross-section Dimensions Size, mm Material properties b h zs lr ' lr Concrete Steel fcm fctm, MPa Ecm, GPa fym, MPa Es, GPa 1 2 3 4 5 6 7 8 9 10 11 B1 Table 3. Basic parameters of the experimental specimens (input data for non-linear analysis) Beam /Slab Series Cross-section Dimensions Size, mm Material properties b h zs lr ' lr Concrete Steel fcm fctm, MPa Ecm, GPa fym, MPa Es, GPa 1 2 3 4 5 6 7 8 9 10 11 B1 150 260 236 0,64 0,91 34,4 4,51 32 440 200 B2 0,91 0,64 32,4 3,95 31,9 435 B3 0,91 0,64 33,9 4,41 33,5 433 B4 250 210 0,66 0,46 28 2,5 28 520 (Ø10) 580 (Ø12) B5 29 2,64 B6 200 300 265 0,85 0,85 26,6 28,3 511 B7 150 250 170 1,28 1,28 74,2 48,3 412 B8 250 210 0,46 1,17 25 28 445 B9 375 120 68 0,64 1,85 30,1 31,5 447 B10 180 180 118 0,59 0,59 30,5 31,6 592 B11 59 54,3 550 B12 100 180 150 0,87 0,87 40 41,2 575 B13 900 150 130 1,06 0,83 33 28,9 450 B14, B15 200 140 90 0,66 0,44 26 1,5 28 530 S7 Slabs 2290 105 1143 0,008 0,0086 35,5 33,7 345 S11 100 0,008 0,004 32,3 S12 100 0,008 0,004 35,6 S15 104 0,0079 0,0087 32,3 S16 105 0,008 0,0088 35,6 S17 100 0,0081 0,0088 35,2 S27 105 0,008 0,0087 35,5 S28 107 0,0081 0,0087 35,3 S33 101 0,008 0,0021 35,3 SH11 Cylindrical Shell 1677 5,02 447 0,611 0,611 43,0 2,0 16,4 670,0 201 SH31 5,0 0,895 0,326 Table 4. Estimated values of the global coefficient R Structures Shlune model Allaix model Coefficient of variation, % m R coeff. var., % factors Vm* Vg VRd VR VR0 VRd 0R Rd R Beams, Frames var det 15,7 17,8… 30,6 1,004 1,55… 1,97 5,8 15,7 1,19 1,21 1,44 Slabs var det 17,3 17,8… 30,6 1,03 1,52… 1,87 5,8 6,56 1,19 1,08 1,28 Note: Value of Vm due to material variability in range from 8,6 % (C50/60) to Vm=21 % (C16/20). Further research is need to recommended appropriate values of the model uncertainty for numerical simulation. It should be noted, that for different FEM-programs values of Rd will be different. These values for FEM-program 150 260 236 0,64 0,91 34,4 4,51 32 440 200 B2 0,91 0,64 32,4 3,95 31,9 435 B3 0,91 0,64 33,9 4,41 33,5 433 B4 250 210 0,66 0,46 28 2,5 28 520 (Ø10) 580 (Ø12)B5 29 2,64 B6 200 300 265 0,85 0,85 26,6 28,3 511 B7 150 250 170 1,28 1,28 74,2 48,3 412 B8 250 210 0,46 1,17 25 28 445 B9 375 120 68 0,64 1,85 30,1 31,5 447 B10 180 180 118 0,59 0,59 30,5 31,6 592 B11 59 54,3 550 B12 100 180 150 0,87 0,87 40 41,2 575 B13 900 150 130 1,06 0,83 33 28,9 450 B14, B15 200 140 90 0,66 0,44 26 1,5 28 530 S7 Slabs 2290 105 1143 0,008 0,0086 35,5 33,7 345 S11 100 0,008 0,004 32,3 S12 100 0,008 0,004 35,6 S15 104 0,0079 0,0087 32,3 S16 105 0,008 0,0088 35,6 S17 100 0,0081 0,0088 35,2 S27 105 0,008 0,0087 35,5 S28 107 0,0081 0,0087 35,3 S33 101 0,008 0,0021 35,3 SH11 Cylindrical Shell 1677 5,02 447 0,611 0,611 43,0 2,0 16,4 670,0 201 SH31 5,0 0,895 0,326 Journal of Sustainable Architecture and Civil Engineering 2018/1/22 84 collected from some references and used for assessment of the coefficient variations RVϑ and model uncertainly factor Rdγ . The model uncertainty factor Rdγ takes into account difference be- tween the real behavior of structure and the results of a numerical modeling suitable for specific structure. The real properties of the material and specimens geometry characteristics obtained by testing were used as an input data for nonlinear analysis. The main characteristics of the analyzed test specimens are presented in Tables 2, 3. As it can be seen from the Table 4, the estimated values of coefficient of variations VRd for model uncertainties are much higher than recommended in codes (for example, in fib MC2010, values in range 1,05…1,1). The same results and conclusions were obtained by Schlune et al. (2011). Schlune concluded that model uncertainties of nonlinear analysis are much higher than in standard design based on engineering formulas and are strongly dependent on modes of failure and adopted failure criteria. Reported in Schlune et al. (2011) coefficient of variation due to model uncertainty for bending failure in range 5…30%, for shear 15…64%. Schlune concluded that due to the lack of data, the choice of model uncertainty often depends on engineering judgment and can be subjective. Note, that coefficient of variations Vm due to material uncertainty (variability) has not a fixed value. In the case of concrete, the mean value of the concrete compressive strength for different classes according to EN 1992-2 (2005) is calculated as: fcm=fck+8 MPa (where 8 1, 64 cMPa = σ , which stan- dard deviation 4, 88c MPaσ = ). For fixed value of standard deviation (as a basic characteristic of the production quality control) 4, 5c MPaσ = , coefficient of variation Vm,c of concrete compressive strength will be in range from 8,6 % (C50/60) to 21 % (C16/20) and coefficient of variation for ma- terials Vm will be in range from Vm=10,48 % to 21,84 % (with fixed value of coefficient of variations Vs=6 % for steel). Table 4 Estimated values of the global coefficient Rγ S tr u ct u re s Shlune model Allaix model Coefficient of variation, % mθ Rγ coeff. var., % factors Vm * Vg VRd VR VR0 VRd 0Rγ Rdγ Rγ Beams, Frames var det 15,7 17,8… 30,6 1,004 1,55… 1,97 5,8 15,7 1,19 1,21 1,44 Slabs var det 17,3 17,8… 30,6 1,03 1,52… 1,87 5,8 6,56 1,19 1,08 1,28 Note: Value of Vm due to material variability in range from 8,6 % (C50/60) to Vm=21 % (C16/20). Further research is need to recommended appropriate values of the model uncertainty for nu- merical simulation. It should be noted, that for different FEM-programs values of Rdγ will be dif- ferent. These values for FEM-program should be estimated based on full probabilistic approach, taking into account statistical parameters of the FEM-model uncertainties and consists of in Pro- gram Manual. 85 Journal of Sustainable Architecture and Civil Engineering 2018/1/22 Fig. 2 Some typical examples of the experimental and predicted force- deflection response of the analyzed specimens (see Tables 2, 3 for designation of the specimens) Fig. 3 For estimatiation of the coefficient VR for FEM-model (see with tables 2, 3) should be estimated based on full probabilistic approach, taking into account statistical parameters of the FEM- model uncertainties and consists of in Program Manual. Fig. 2. Some typical examples of the experimental and predicted force-deflection response of the analyzed specimens (see Tables 2, 3 for designation of the specimens) Fig. 3. For estimatiation of the coefficient VR for FEM-model (see with tables 2, 3) 6. Conclusions. should be estimated based on full probabilistic approach, taking into account statistical parameters of the FEM- model uncertainties and consists of in Program Manual. Fig. 2. Some typical examples of the experimental and predicted force-deflection response of the analyzed specimens (see Tables 2, 3 for designation of the specimens) Fig. 3. For estimatiation of the coefficient VR for FEM-model (see with tables 2, 3) 6. Conclusions. should be estimated based on full probabilistic approach, taking into account statistical parameters of the FEM- model uncertainties and consists of in Program Manual. Fig. 2. Some typical examples of the experimental and predicted force-deflection response of the analyzed specimens (see Tables 2, 3 for designation of the specimens) Fig. 3. For estimatiation of the coefficient VR for FEM-model (see with tables 2, 3) 6. Conclusions. Safety format suitable for nonlinear analysis (pseudo-static response) that based on global resis- tance in accordance with fib MC2010 concept are presented. The following conclusions can be adopt: (1) the differences between proposed methods are not significant; (2) fixed value of global safety factor 1, 27Rγ = in accordance with fib MC2010 (2010) and EN 1992-2 (2005) is not good approach for safety assessment and sometimes can be uncon- servative results; (3) the values of the global resistance factor Rγ should be estimated separately for different computer programs, which are used for non-linear analysis (pseudo-static response Conclusions Journal of Sustainable Architecture and Civil Engineering 2018/1/22 86 of the structural system), based on experimental results. These values for separate computer programs should be estimated based on full probabilistic approach, taking into account statistical parameters of the FEM-model uncertainties. 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Vlassis, A. G. 2007. Progressive collapse assess- ment of tall buildings: Doctoral dissertation, Impe- rial College London, University of London. Yi, W. J.; He, Q. F.; Xiao, Y.; Kunnath, S. K. 2008. Expe- rimental study on progressive collapse-resistant behaviour of reinforced concrete frame structures, ACI Structural Journal 105(4): 433. About the Authors ANDREI TUR Assoc. Professor, PhD, engineer Brest State Technical University, Faculty of Civil Engineering, Department “Building constructions” Main research area Robustness, nonlinear static and dynamic analysis of structures, reliability of building structures Address 267, Moskovskaya str., 224017, Brest, Belarus Tel. +375 29 5253206 E-mail: aturphd@gmail.com VICTAR TUR Professor, DSc, PhD Brest State Technical University, Faculty of Civil Engineering, Department “Technology of concrete and building materials” Main research area Self-stressing concrete structures, reliability of building structures Address 267, Moskovskaya str., 224017, Brest, Belarus Tel. +375 33 6729404 E-mail: profturvic@gmail.com