Reliability Approaches to Modeling of the Nonlinear Pseudo-static Response of RC-structural Systems in Accidental Design Situations

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.


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.
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 FEMmethods 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 or checking of the existing structures.ysis (static and dynamic) is most widely used as a main computational tool for checking of tructural systems in accidental design situations (Accidental Limit States Checking).d in Červenka (2013a), "evaluations of the nonlinear analysis are supported by rapid increase of er as well as new capabilities of the available tools for numerical simulations of structural hed works dealing with nonlinear finite element analysis of concrete systems emerged in the late s focused on various aspects of element formations, including crack propagation and the bonding n general case two basic FEM-methods are used for non-linear modeling: 1) so-called stiffness Stiffness Model); 2) Layered Model.daptation analysis is purposed to be an alternative for a full nonlinear analysis (Layered Model) distributions, deformations, crack patterns and crack-width in reinforced concrete structures.ptation analysis both standards linear elastic material as well as non-linear material behavior can d Schnobrich 1991).Nonlinear materials can be defined through a uni-axial stress-strain curve, and in the compressive state.Nonlinear stress-strain curves may also be defined for bar and grid Layered model approach each concrete layer is assumed to be in a state of plane stress and the ution of the concrete section is modeled by a piecewise constant approximation., for the reinforcement concrete section, the final form of the stress resultant constitutive matrix at t can be written as: is the stiffness matrix that can be established by assembling the contributions of all the concrete ers and transverse shear stiffness.wn above, nonlinear analysis take into account the nonlinear deformation properties of RCphysical constitutive relations (" σ ε  " for material properties) and makes possible a simulation behavior.If reflects an integral response, where all local sections interact and therefore it requires ach for safety verification (note, that in partial safety factor (PSF) method (EN 1990(EN : 2006) ) we robabilities of separate materials, but do not evaluate the failure probability on the structural underlined, that nonlinear analysis offers a verification of global resistance and requires a safety resistance (Červenka 2013b).In accordance with Červenka (2013b), the term global resistance sed for "assessment of structural response on higher structural level than a cross-section".The nce is introduced in Červenka (2013b) in order to distinguish the newly introduced check of evel, as compared to local safety check in the partial safety factor method (PSF-method) in N 1990 (2006). (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 (" 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(EN : 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 1990with EN (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: nlinear analysis in accordance with actual codes provision.from CEM MC78 to fib MC2010) of the non-linear safety format development was rgio (2015).n of the new fib MC2010 (2010), a different perspective was placed on nonlinear analysis design condition to be used in safety format for nonlinear analysis is written in the g internal forces domain: n value of the action; d R  is the design value of resistance.hes are proposed to evaluate the design resistance d R (depending on various levels of listic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) d; and (3) the partial factor method (PSF-method).format, the resistance is considered on a global structural level.Two alternative methods 010 (2010) for the derivation of the design resistance d R : (1) global resistance factor ed from EN 1992EN -2 (2005)), slightly modified); and (2) ECOV-method, proposed by ra and Holicky (2011) (estimations of coefficient variation for resistance).argin can be expressed by the global safety factor as:

R
 cover all uncertainties and can be related to the coefficient of variations of resistance ution (!) according EN 1992EN -2 (2005))) as exp( ) (2) where: d E -is the design value of the action; d R -is the design value of resistance.
Three different approaches are proposed to evaluate the design resistance d R (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 d R : (1) global resistance factor method (which was adopted from EN 1992EN -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: near analysis in accordance with actual codes provision.m CEM MC78 to fib MC2010) of the non-linear safety format development was io (2015).f the new fib MC2010 (2010), a different perspective was placed on nonlinear analysis esign condition to be used in safety format for nonlinear analysis is written in the nternal forces domain: value of the action; d R  is the design value of resistance.s are proposed to evaluate the design resistance d R (depending on various levels of tic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) nd (3) the partial factor method (PSF-method).mat, the resistance is considered on a global structural level.Two alternative methods (2010) for the derivation of the design resistance d R : (1) global resistance factor from EN 1992EN -2 (2005)), slightly modified); and (2) ECOV-method, proposed by and Holicky (2011) (estimations of coefficient variation for resistance).gin can be expressed by the global safety factor as: cover all uncertainties and can be related to the coefficient of variations of resistance ion (!) according EN 1992-2 ( 2005)) as exp( ) , where in denominator of the right hand side in eq.

R m Rd
     (Sykora and Holicky 2011).The first factor m  is related to material (3) where: m R -is the mean resistance.
The global safety factor R γ cover all uncertainties and can be related to the coefficient of variations 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 γ = γ ⋅ γ (Sykora and Holicky 2011).The first factor m γ is related to material uncertainty and can be established by probabilistic analysis.The second factor Rd γ is related to model and geometrical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992EN -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 d R is then derived by division of the obtained mean resistance m R by global resistance factor R γ : mentations of probabilistic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) obal resistance method; and (3) the partial factor method (PSF-method).the global resistance format, the resistance is considered on a global structural level.Two alternative methods entioned in fib MC2010 (2010) for the derivation of the design resistance d R : (1) global resistance factor d (which was adopted from EN 1992EN -2 (2005)), slightly modified); and (2) ECOV-method, proposed by nka (2013b) and Sykora and Holicky (2011) (estimations of coefficient variation for resistance).this case, the safety margin can be expressed by the global safety factor as: ere: m R  is the mean resistance.e global safety factor R  cover all uncertainties and can be related to the coefficient of variations of resistance ccording a LN-distribution (!) according EN 1992EN -2 (2005))) as exp( ) . simplified formulation was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq.product of two factors R m Rd      (Sykora and Holicky 2011).The first factor m  is related to material ainty and can be established by probabilistic analysis.The second factor Rd  is related to model and trical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2 )).
it was stated in Sangiorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, ew contributions were found in literature (Schlune et al. 2011, Allaix et al. 2013).e first contribution was presented by Schlune et al. (2011).Design resistance d R is then derived by division obtained mean resistance m R by global resistance factor R  : ( , , )  again based on the assumption of a lognormal distributed resistance.
odel uncertainties are explicitly taken into accounts thought the use of the bias factor θ m , which is defined as ean ratio of experimental to predicted resistance (in accordance with Schlune et al. (2011)) its value varies en 0.7 and 1.2 for failure in compression, bending and shear).The coefficient variation of structural resistance written as follows: , , ( , , ) this case, the global resistance factor R  is derived from coefficient of variations of the structural resistance stimated by probabilistic method or based on Červenka method ECOV): exp( ) ere:  is the reliability index in accordance with EN 1990; R  -is the sensitivity factor for resistance. (4) where: -again based on the assumption of a lognormal distributed resistance.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 R V is written as follows: ree different approaches are proposed to evaluate the design resistance d R (depending on various levels of entations of probabilistic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) bal resistance method; and (3) the partial factor method (PSF-method).the global resistance format, the resistance is considered on a global structural level.Two alternative methods ntioned in fib MC2010 (2010) for the derivation of the design resistance d R : (1) global resistance factor d (which was adopted from EN 1992EN -2 (2005)), slightly modified); and (2) ECOV-method, proposed by ka (2013b) and Sykora and Holicky (2011) (estimations of coefficient variation for resistance).this case, the safety margin can be expressed by the global safety factor as: ere: m R  is the mean resistance.e global safety factor R  cover all uncertainties and can be related to the coefficient of variations of resistance cording a LN-distribution (!) according EN 1992EN -2 (2005))) as exp( ) . implified formulation was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq.product of two factors R m Rd      (Sykora and Holicky 2011).The first factor m  is related to material ainty and can be established by probabilistic analysis.The second factor Rd  is related to model and trical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2 ). it was stated in Sangiorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, w contributions were found in literature (Schlune et al. 2011, Allaix et al. 2013).e first contribution was presented by Schlune et al. (2011).Design resistance d R is then derived by division obtained mean resistance m R by global resistance factor R  : ( , , )  again based on the assumption of a lognormal distributed resistance.
del uncertainties are explicitly taken into accounts thought the use of the bias factor θ m , which is defined as an ratio of experimental to predicted resistance (in accordance with Schlune et al. (2011)) its value varies n 0.7 and 1.2 for failure in compression, bending and shear).The coefficient variation of structural resistance written as follows: ere: , , ( , , ) this case, the global resistance factor R  is derived from coefficient of variations of the structural resistance timated by probabilistic method or based on Červenka method ECOV): exp( ) ere:  is the reliability index in accordance with EN 1990; R  -is the sensitivity factor for resistance. ( where: , , V 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 andMancini 2007, Allaix et al. 2013) the design resistance d R is derived by divisions of obtained resistance factor R γ and the model uncertainty factor Rd γ : where: d E  is the design value of the action; d R  is the design value of resistance.Three different approaches are proposed to evaluate the design resistance d R (depending on various levels of lementations of probabilistic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) 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 mentioned in fib MC2010 (2010) for the derivation of the design resistance d R : (1) global resistance factor hod (which was adopted from EN 1992EN -2 (2005)), slightly modified); and (2) ECOV-method, proposed by venka (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: where: m R  is the mean resistance.The global safety factor R  cover all uncertainties and can be related to the coefficient of variations of resistance (according a LN-distribution (!) according EN 1992EN -2 (2005))) as exp( )  is related to model and metrical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2 5)).
As it was stated in Sangiorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, 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 d R is then derived by division e obtained mean resistance m R by global resistance factor R  : ( , , ) where: exp( )  again based on the assumption of a lognormal distributed resistance.
Model uncertainties are explicitly taken into accounts thought the use of the bias factor θ m , which is defined as mean ratio of experimental to predicted resistance (in accordance with Schlune et al. ( 2011)) its value varies een 0.7 and 1.2 for failure in compression, bending and shear).The coefficient variation of structural resistance is written as follows: where: , , ( , , ) In this case, the global resistance factor R  is derived from coefficient of variations of the structural resistance (estimated by probabilistic method or based on Červenka method ECOV): exp( ) 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 R γ is derived from coefficient of variations of the structural resistance R V (estimated by probabilistic method or based on Červenka method ECOV): ty assessment.The design condition to be used in safety format for nonlinear analysis is written in the actions and resisting internal forces domain: e: d E  is the design value of the action; d R  is the design value of resistance.e different approaches are proposed to evaluate the design resistance d R (depending on various levels of ntations of probabilistic theory): (1) full probabilistic method, recommended by JCSS as a basic method; (2) al resistance method; and (3) the partial factor method (PSF-method).e global resistance format, the resistance is considered on a global structural level.Two alternative methods tioned in fib MC2010 (2010) for the derivation of the design resistance d R : (1) global resistance factor (which was adopted from EN 1992EN -2 (2005)), slightly modified); and (2) ECOV-method, proposed by a (2013b) and Sykora and Holicky (2011) (estimations of coefficient variation for resistance).is case, the safety margin can be expressed by the global safety factor as: e: m R  is the mean resistance.global safety factor R  cover all uncertainties and can be related to the coefficient of variations of resistance ording a LN-distribution (!) according EN 1992EN -2 (2005))) as exp( )  is related to model and ical uncertainties and recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2 was stated in Sangiorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, contributions were found in literature (Schlune et al. 2011, Allaix et al. 2013).first contribution was presented by Schlune et al. (2011).Design resistance d R is then derived by division tained mean resistance m R by global resistance factor R  : ( , , )  again based on the assumption of a lognormal distributed resistance.
el uncertainties are explicitly taken into accounts thought the use of the bias factor θ m , which is defined as n ratio of experimental to predicted resistance (in accordance with Schlune et al. ( 2011)) its value varies 0.7 and 1.2 for failure in compression, bending and shear).The coefficient variation of structural resistance ritten as follows: e: , , ( , , ) is case, the global resistance factor R  is derived from coefficient of variations of the structural resistance mated by probabilistic method or based on Červenka method ECOV): exp( ) e:  is the reliability index in accordance with EN 1990; R  -is the sensitivity factor for resistance. ( where: β is the reliability index in accordance with EN 1990; R α -is the sensitivity factor for resistance.exp( )

The model uncertainty factor
 is the sensitivity factor for resistance model uncertainty ( R   <1 in order to account for safety assessment of resistance); R V   is the coefficient of variations of the resistance model uncertainly.value of this coefficient of variations can be obtained based on experimental results according to EN 1990 checking of the RC-structural system in accidental design situation, two main issues must be solved: (1) to e the pseudo-static response of the modified structural system under accidental loads; (2) to determined alue of the reliability index for accidental design (required level of reliability).
seudo-static response of the structural system with a removed vertical load bearing elements.(8) where: -is the sensitivity factor for resistance model uncertainty ( R α  <1 in order to account for separate safety assessment of resistance); R V ϑ -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 1990to EN (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).
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 re-Pseudostatic 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) sponse, that taking into account inertia effects, caused by suddenly applied gravity load.
In general case, based on energetic consideration (see Fig. 1): with removed vertical load bearing elements.If a dynamic increase factor (DIF) is known, for deformation-lled actions, the resulting deformations are compared to the expected deformation capacities; for the force lled action, the member strength is not modified and shell not be less than the maximum internal member (demands).Otherwise, calculation procedure based on the energetic approach should be used.The basic ions of this procedure are described in detail in Tur (2012).The purpose here is to analyze the structural se of RC-structural systems subjected to a sudden column loss.e procedure, which is used for obtaining of the pseudo-static nonlinear response of the structural system, ts of the following main steps: (1) Calculate the static non-linear response " F δ  " for the modified structural with a removed vertical load bearing element according to certain rules (Tur 2012, Vlassis 2009) (see Fig. 1,  ); (2) Calculate the pseudo-static response, that taking into account inertia effects, caused by suddenly applied y load.general case, based on energetic consideration (see Fig. 1): , 0 ( ) Pseudo-static response is equal: To assessment of the pseudo-static response of the structural system in accordance with (Tur 2012, Vlassis 2009) do-static response is equal: , 0 1 ( ) equired level of reliability for accidental design situation.eneral case, the probability of structure collapse due to postulated abnormal event can be written as: ( ) ( ) ( ) ( ) as shown in Ellingwood (2002), in a "specific local resistance" design strategy, the focus is on minimizing lity ( ) i P F DH , that is, to minimize the likelihood of initiation of damage that may lead to progressive .strategy may be difficult or uneconomical, and may leave some significant hazards unaddressed.ordingly, it is likely that ( ) i P D H will very close to 1,0 in many practical cases, meaning that the collapse lity becomes, approximately: ( ) ( ) ( ) in minimizing the conditional probability ( ) i P F H , that the science and art of the structural engineer s paramount (Ellingwood 2002).ay be assumed that the occurrence of the abnormal event i H can be modeled as a Poisson process with ean rate of occurrence i  .The probability of occurrence of this abnormal event during some reference T, is thus approximately ( ) (for very small i  ) (Ellingwood 2002).In the case of fire, gas n and some other accidental loads, parameter i  may be related to building floor area ( odel), of the structural system based on principles of mechanics and supplemented, where possible, with ental data (!).The load and resistance variables are expressed by vector X.We must then determine the lity distribution of each variable and integrate the joint density function of X over that region of probability here G(X) <0 to compute in accordance with EN 1990 conventional limit state probability.But, we must to er that it is very difficult and complex way (especially for structural systems).rnatively, FORM -analysis may be used to compute a conditional reliability index  defined as: re: G  and G   is mean and standard deviation of G(X).
ording to Ellingwood (2002), the reliability index is related to ( ) i P F DH through: - is the percent-point function of the standard Normal probability distribution.
( ) 14) can be rewritten as: - as shown in Ellingwood (2002), the first-generation probability-based Limit State Design Criteria (such as, ple, EUROCODES) all are based, to varying degrees, on reliability of individual structural members and ents.ever, to implement reliability-based design criteria against progressive collapse in practice sense, the limit (10) In general case, the probability of structure collapse due to postulated abnormal event can be written as: To assessment of the pseudo-static response of the structural system in accordance with (Tur 2012, Vlassis 2009) Pseudo-static response is equal: 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: ( ) ( ) ( ) ( ) As was shown in Ellingwood (2002), in a "specific local resistance" design strategy, the focus is on minimizing probability ( ) i P 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 ( ) i P D H will very close to 1,0 in many practical cases, meaning that the collapse probability becomes, approximately: ( ) ( ) ( ) It is in minimizing the conditional probability ( ) i P 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 i H 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 ( ) (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 f p A    , in which f A -floor area and 1 2 p p p   , where term 1 p  probability of occurrence of hazard per unit area and 2 1.0 p  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)  (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: where: G  and G   is mean and standard deviation of G(X).
According to Ellingwood (2002), the reliability index is related to ( ) i P F DH through: - is the percent-point function of the standard Normal probability distribution.
With ( )   i i P H T   , eq. ( 14) can be rewritten as: - As was shown in Ellingwood (2002), the first-generation probability-based Limit State Design Criteria (such as, As was shown in Ellingwood (2002), in a "specific local resistance" design strategy, the focus is on minimizing probability ( ) i P 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 ( ) i P D H will very close to 1,0 in many practical cases, meaning that the collapse probability becomes, approximately: To assessment of the pseudo-static response of the structural system in accordance with (Tur 2012, Vlassis 2009) eudo-static response is equal: Required level of reliability for accidental design situation.general case, the probability of structure collapse due to postulated abnormal event can be written as: s was shown in Ellingwood (2002), in a "specific local resistance" design strategy, the focus is on minimizing bility ( ) i P F DH , that is, to minimize the likelihood of initiation of damage that may lead to progressive se. is strategy may be difficult or uneconomical, and may leave some significant hazards unaddressed.ccordingly, it is likely that ( ) i P D H will very close to 1,0 in many practical cases, meaning that the collapse bility becomes, approximately: ( ) ( ) ( ) is in minimizing the conditional probability ( ) i P F H , that the science and art of the structural engineer es paramount (Ellingwood 2002).may be assumed that the occurrence of the abnormal event i H can be modeled as a Poisson process with mean rate of occurrence i  .The probability of occurrence of this abnormal event during some reference T, is thus approximately ( ) (for very small i  ) (Ellingwood 2002).In the case of fire, gas sion and some other accidental loads, parameter i  may be related to building floor area ( i model), of the structural system based on principles of mechanics and supplemented, where possible, with imental data (!).The load and resistance variables are expressed by vector X.We must then determine the bility distribution of each variable and integrate the joint density function of X over that region of probability where G(X) <0 to compute in accordance with EN 1990 conventional limit state probability.But, we must to ber that it is very difficult and complex way (especially for structural systems).lternatively, FORM -analysis may be used to compute a conditional reliability index  defined as: here: G  and G   is mean and standard deviation of G(X).
ccording to Ellingwood (2002), the reliability index is related to ( ) i P F DH through: - is the percent-point function of the standard Normal probability distribution. ( It is in minimizing the conditional probability ( ) i P 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 i H can be modeled as a Poisson process with yearly mean rate of occurrence i λ .The probability of occurrence of this abnormal event during some reference period T, is thus approximately ( ) i i P H T = λ (for very small i λ ) (Ellingwood 2002).In the case of fire, gas explosion and some other accidental loads, parameter p p p = ⋅ , where term 1 p -probability of occurrence of hazard per unit area and 2 1.0 p < represents effect of warning and control systems).
As it was shown in (Ellingwood 2002), to evaluate ( ) i P 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

Required level of reliability for accidental design situation
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: are approximately: g): 2x10 -5 /yr; ling): 2x10 -6 /yr; elling): 6x10 -4 /yr; ding): 5x10 -8 /yr.ood 2002), to evaluate ( ) i P F DH , one must postulate a mathematical model, G(X) system based on principles of mechanics and supplemented, where possible, with d and resistance variables are expressed by vector X.We must then determine the variable and integrate the joint density function of X over that region of probability te in accordance with EN 1990 conventional limit state probability.But, we must to lt and complex way (especially for structural systems).ysis may be used to compute a conditional reliability index  defined as: n and standard deviation of G(X).
002), the reliability index is related to ( ) i P F DH through: - the percent-point function of the standard Normal probability distribution.can be rewritten as: - d ( 2002), the first-generation probability-based Limit State Design Criteria (such as, ll are based, to varying degrees, on reliability of individual structural members and iability-based design criteria against progressive collapse in practice sense, the limit ndex) must be evaluated for a structural system (!).In contrast to member reliability, (13) where: G µ and G σ -is mean and standard deviation of G(X).
According to Ellingwood (2002), the reliability index is related to ( ) i P F DH through: rrence for gas explosions, bomb explosions and vehicular collisions in accordance with 1991) are approximately: dwelling): 2x10 -5 /yr; r dwelling): 2x10 -6 /yr; (per dwelling): 6x10 -4 /yr; per building): 5x10 -8 /yr.Ellingwood 2002), to evaluate ( ) i P F DH , one must postulate a mathematical model, G(X) uctural system based on principles of mechanics and supplemented, where possible, with he load and resistance variables are expressed by vector X.We must then determine the of each variable and integrate the joint density function of X over that region of probability compute in accordance with EN 1990 conventional limit state probability.But, we must to difficult and complex way (especially for structural systems).
-analysis may be used to compute a conditional reliability index  defined as: is mean and standard deviation of G(X).
ood ( 2002), the reliability index is related to ( ) i P F DH through: - )] i H is the percent-point function of the standard Normal probability distribution.
eq. ( 14) can be rewritten as: - ingwood ( 2002), the first-generation probability-based Limit State Design Criteria (such as, DES) all are based, to varying degrees, on reliability of individual structural members and ent reliability-based design criteria against progressive collapse in practice sense, the limit ability index) must be evaluated for a structural system (!).In contrast to member reliability, ( is the percent-point function of the standard Normal probability distribution.With ( ) 14) can be rewritten as: rence for gas explosions, bomb explosions and vehicular collisions in accordance with 1991) are approximately: welling): 2x10 -5 /yr; r dwelling): 2x10 -6 /yr; per dwelling): 6x10 -4 /yr; er building): 5x10 -8 /yr.llingwood 2002), to evaluate ( ) i P F DH , one must postulate a mathematical model, G(X) ctural system based on principles of mechanics and supplemented, where possible, with he load and resistance variables are expressed by vector X.We must then determine the f each variable and integrate the joint density function of X over that region of probability compute in accordance with EN 1990 conventional limit state probability.But, we must to ifficult and complex way (especially for structural systems).
-analysis may be used to compute a conditional reliability index  defined as: is mean and standard deviation of G(X).
ood ( 2002), the reliability index is related to ( ) i P F DH through: - )] i is the percent-point function of the standard Normal probability distribution.
q. ( 14) can be rewritten as: - ngwood ( 2002), the first-generation probability-based Limit State Design Criteria (such as, ES) all are based, to varying degrees, on reliability of individual structural members and nt reliability-based design criteria against progressive collapse in practice sense, the limit bility index) must be evaluated for a structural system (!).In contrast to member reliability, 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, 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 magnitude less, depending on the redundancy in the system and the degree continuity between members.
For example, if 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 β 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 R γ 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 V R , can be estimated from mean R m and characteristic R k values of resistance (pseudo-static response of the structural system).In this case, coefficient variations of resistance V R can be obtained from following equation: !) even at the present state of art and with computational resources available (Ellingwood sis of a damaged structure can be performed, an acceptable value of  upon which to limit states is suggested by eq. ( 15).od (2002), the probability of structural system failure is an order of magnitude less, cy in the system and the degree continuity between members.6  to 10 -5 , than the conditional failure probability for the structural system should be on the target value of reliability index tag  should be the order of 1,5.Load and resistance be consistent with the reliability.
lobal resistance and global safety factors for pseudo-static response.lysis the value of the global resistance factor R  was defined in accordance with (Sykora q. (7).As it was shown above, the ECOV-method is based on idea that the random hich is again described by the coefficient variation VR, can be estimated from mean Rm s of resistance (pseudo-static response of the structural system).In this case, coefficient can be obtained from following equation: mean and characteristic values of resistance (pseudo-static response, as was shown in o separate non-linear analysis using mean and characteristic values of input material linear analysis of the statically undetermined an encastre RC-beam and values of the d global resistance coefficient R  obtained by calculations are presented in Table 1. where: , m k R 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 V R and global resistance coefficient R γ 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 pseudostatic response 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 andHolicky 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 R  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 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.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 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.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.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.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: f =f +8 MPa (where 8 1, 64 MPa   , which standard deviation 4,88 MPa   ).For fixed

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.At the second stage of analysis the coefficient of variations R V ϑ of the computer model uncertainties was assessed based on theoretical background described in Annex D (EN 1990(EN : 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 R V ϑ and model uncertainly factor Rd γ .The model uncertainty factor Rd γ 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 V Rd 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 V m 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 1992EN -2 (2005) )  ).For fixed value of standard deviation (as a basic characteristic of the production quality control) 4,5 , coefficient of variation V m,c of concrete compressive strength will be in range from 8,6 % (C50/60) to 21 % (C16/20) and coefficient of variation for materials V m will be in range from V m =10,48 % to 21,84 % (with fixed value of coefficient of variations V s =6 % for steel).Safety format suitable for nonlinear analysis (pseudo-static response) that based on global resistance 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, 27 R γ = in accordance with fib MC2010 (2010) and EN 1992EN -2 (2005) ) is not good approach for safety assessment and sometimes can be unconservative results; (3) the values of the global resistance factor R γ should be estimated separately for different computer programs, which are used for non-linear analysis (pseudo-static response Conclusions was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq.rs R m Rd      (Sykora and Holicky 2011).The first factor m  is related to material tablished by probabilistic analysis.The second factor Rd  is related to model and nd recommended value are in range 1.05…1.1 only (!) (as suggested by EN 1992-2 iorgio (2015), after the new fib MC 2010 (2010), although the topic is still controversial, found in literature (Schlune et al. 2011, Allaix et al. 2013).as presented by Schlune et al. (2011).Design resistance d R is then derived by division nce R by global resistance factor  : formulation was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq. is product of two factors R m Rd      (Sykora and Holicky 2011).The first factor m  is related to material ertainty and can be established by probabilistic analysis.The second factor Rd are the coefficients of variations of the geometrical, model and material uncertainties ectively, estimated in accordance withSchlune et al. (2011).According with the second contributions, proposed by(Allaix andMancini 2007, Allaix et al. 2013) the design stance d R is derived by divisions of obtained resistance factor R  and the model uncertainty factor Rd  : formulation was proposed in fib MC2010 (2010), where in denominator of the right hand side in eq.oduct of two factors R m Rd      (Sykora and Holicky 2011).The first factor m  is related to material nty and can be established by probabilistic analysis.The second factor Rd Rd γ 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 γ can be derived using the following expression from Schlune et al. (2011): model uncertainty factor Rd  takes into account the difference between the real behavior of the structure and ults obtained based on a numerical model.The model uncertainty factor Rd  can be derived using the ng expression from Schlune et al. (2011):

Fig
Fig. 1.To assessment of the pseudo-static response of the structural system in accordance with(Tur 2012, Vlassis 2009)

Fig
Fig. 1.To assessment of the pseudo-static response of the structural system in accordance with(Tur 2012, Vlassis 2009) , S12, S15, S16, S17, S27, S28, S33Cardenas 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 fixed value of standard deviation (as a basic characteristic of the production quality control) , S12, S15, S16, S17, S27, S28, S33Cardenas 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 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 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) Fig. 2Some typical examples of the experimental and predicted forcedeflection response of the analyzed specimens (see Tables2, 3for designation of the specimens) For estimatiation of the coefficient V R for FEM-model (see with tables 2, 3) should be estimated based on full probabilistic approach, taking into account statistical parameters of the FEMmodel uncertainties and consists of in Program Manual.

Fig. 2 .Fig. 3 .
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. 2 .Fig. 3 .
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. 2 .Fig. 3 .
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)

Table
Loading arrangement for experimental specimens

Table 2 .
Loading arrangement for experimental specimens

Table 2 .
Loading arrangement for experimental specimens

Table 2 .
Loading arrangement for experimental specimens

Table 2 .
Loading arrangement for experimental specimens

Table 2 .
Loading arrangement for experimental specimens

Table 3 .
Basic parameters of the experimental specimens (input data for non-linear analysis)

Table 4 .
Estimated values of the global coefficient R  m  is calculated as: f cm =f ck +8 MPa (where 8 1, 64 c c MPa σ =

Table 4
Value of V m due to material variability in range from 8,6 % (C50/60) to V m =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 γ will be different.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 Program Manual. Note: