4.2 Partial factor method based on the design value approach

Transcription

1 4.2 Partial factor method based on the design value approach 2 nd draft Milan Holicky & Miroslav Sykora* * Czech Technical University in Prague, Klokner Institute, Prague, Czech Republic milan.holicky@klok.cvut.cz; miroslav.sykora@klok.cvut.cz General formulation This section describes the application of the partial factor method (recommended in the Eurocodes for new structures) for the assessment of existing structures, using the design value approach conforming to ISO 2394 (1998) and EN 1990 (2002). Partial factors derived in this section are intended to be applied in conjunction with load combination rules (6.10), or (6.10a,b) given in EN 1990 (2002). In case of structures without prestressing, the reliability verification format may be written as: R d E d = Σ j γ G,j G k,j + γ Q,1 Q k,1 + Σ i γ Q,i ψ 0,i Q k,i ; j 1, i > 1 (4.2-1) or, alternatively, the less favourable of the two following expressions: R d E d = Σ j γ G,j G k,j + Σ i γ Q,i ψ 0,i Q k,i ; j 1, i 1 R d E d = Σ j ξ j γ G,j G k,j + γ Q,1 Q k,1 + Σ i γ Q,i ψ 0,i Q k,i ; j 1, i > 1 (4.2-2a) (4.2-2b) where R d (f ck / γ C, f yk / γ S, ) is the design value of resistance, + implies to be combined with, Σ the combined effect of and ξ is a reduction factor for unfavourable permanent actions G. In order to apply the design value approach elaborated here, the characteristic values shall be assessed from actual distributions of basic material properties and actions (based on prior information, or results of tests or the combination of both - see chapter???). The partial factors for material properties and actions can be assessed using procedures described in this section. The values for the factors ξ and ψ 0 provided in EN 1990 (2002) are also accepted for the assessment of existing structures. 1 Last modified:

2 fib SAG7 - group A.2 - draft of chapter 4.2 (Holicky & Sykora) Material factors The design value R d of the resistance variable R is defined by the relationship (4.1-12) 1. The partial factor given by (4.1-10) may be assumed as a product of: γ M = γ Rd γ m = γ Rd1 γ Rd2 γ m (4.2-3) where γ Rd1 denotes the partial factor accounting for model uncertainty, γ Rd2 2 is the partial factor accounting for geometrical uncertainties and γ m is the reliability-based partial factor account- may be as- ing for variability of the material and statistical uncertainty. Values of γ Rd1 = 1.05 for concrete strength and γ Rd1 = for reinforcement sumed in common cases. However, larger model uncertainty may need to be considered for instance in the case of punching shear where concrete crushing is governing. A value of γ Rd2 = 1.05 may be assumed for geometrical uncertainties of the concrete section size or reinforcement position. When relevant measurements on an existing structure indicate insignificant variability of geometrical properties, γ Rd2 = 1.0 may be considered. In case the design value method is applied, the partial factor γ m and the characteristic value shall be derived from the actual distribution of the material property under consideration (based on prior information, or results of tests or the combination of both - see chapter???). Assuming a lognormal distribution for the material property, variation of the partial factor γ m obtained from Eq. (4.1-21) with the coefficient of variation δ m is shown in Figure for α R = 0.8 and target reliabilities β = 2.3, 3.1, 3.8 or 4.3 (very low, low, medium and high fail- ure consequences in ULS, respectively, ISO (2003)). Figure 4.2-1: Variation of the partial factor γ m with the coefficient of variation δ m for α R = 0.8 and β = 2.3, 3.1, 3.8 or Referring to the 1 st draft of section Last modified:

3 As an example, having measurements of geometrical properties and material properties yielding coefficients of variation of concrete strength and reinforcement δ c = 0.15 and δ s = 0.05, the following partial factors are obtained: for β = 3.8: γ C = = 1.29; γ S = = 1.10 for β = 3.1: γ S = = 1.19; γ S = = 1.07 (4.2-4) Note that the partial factor γ C = 1.5 provided in EN (2004) has been derived considering γ Rd1 = 1.05, γ Rd2 = 1.05, δ c = 0.15, β = 3.8 and the additional uncertainty due to the fact that in structural design the concrete strength is assessed from samples not taken from a structure (expressed by a conversion factor η 1.1). The partial factor γ S = 1.15 provided in EN (2004) has been derived considering γ Rd1 = 1.025, γ Rd2 = 1.05, δ s = 0.06 and β = Permanent action factor The design value G d of the permanent action effect G is defined by the general relationship (4.1-13). The partial factor given by (4.1-9) may be obtained as follows: γ G = γ Sd,g γ g (4.2-5) where γ Sd,g denotes the partial factor accounting for model uncertainty and γ g is the reliabilitybased partial factor accounting for variability of the permanent action and the statistical uncertainty. γ Sd,g = 1.05 is normally assumed in structural design. However, this value may be reduced to γ Sd,g = 1.0 when: the model of the permanent action effect is based on measurements and transformation from permanent action to its effect does not yield any significant uncertainties and there is no indication that the permanent action will considerably change in future. In order to apply the design value method, the partial factor γ G and the characteristic value shall be derived from the actual distribution of the permanent action under consideration (preferably based on measurements). Assuming a normal distribution for the permanent action, variation of the partial factor γ g obtained from Eq. (4.1-22) with the coefficient of variation δ g is shown in Figure for α E = -0.7 and the target reliabilities β = 2.3, 3.1, 3.8 or Last modified:

4 fib SAG7 - group A.2 - draft of chapter 4.2 (Holicky & Sykora) Figure 4.2-2: Variation of the partial factor γ g with the coefficient of variation δ g for α E = -0.7 and β = 2.3, 3.1, 3.8 or 4.3. As an example, assuming γ Sd d,g = 1.0 and coefficients of variation for self-weight δ g0 = 0.05 and other permanent loads δ g1 = 0.1, the following partial factors are obtained: for β = 3.8: γ G0 = = 1.13; γ G1 = = 1.27 for β = 3.1: γ G0 = = 1.11; γ G1 = = 1.22 (4.2-6) Note that the partial factor γ G = 1.35 given in EN 1990 (2002) has been derived considering γ Sd = 1.05, δ g = 0.1 and β = Variable action factors In many cases no additional information on variable loads, except that provided by valid codes for structural design, is available in assessments of existing structures. The characteris- of tic values and partial factors for variable loads should then be based on recommendations such codes. However, if applicable partial factors should be adjusted to a specific situation of the existing structure with respect to assumed reliability levels, remaining working lives, etc. When site- or structure-specific data on variable loads can be gathered and a detailed assessusing the procedure ment needs to be made, partial factors for variable loads may be derived described in this section. The partial factor for the variable load effect Q related to the target reliability level β and reference period t ref may be obtained as follows: γ Q = F -1 Q,tref [Φ(-α E β), t ref ] / Q k (4.2-7) where F -1 Q,tref = inverse cumulative distribution function of maxima of the variable load effect during the reference period t re ef. The distribution of the load maxima should be based on the same reference period t ref as used for the reliability index β, EN 1990 (2002). 4 Last modified:

5 In general the remaining working life t r may differ from the reference period t ref. Nevertheless, for existing structures exposed to deterioration, it is mostly required to consider t ref equal to t r. Note that the target reliability index β may be specified independently of the remaining working life, Holický & Sýkora (2011). In common cases EN 1990 (2002) allows to approximate the sensitivity factor α E by the value -0.7 for the leading variable action and by for an accompanying variable action. However, available measurements may often lead to reduction of uncertainties related to resistance and permanent action effect when assessing existing structures. Then, the sensitivity factors for the resistance and permanent actions decrease and the absolute values of the sensitivity factor for the variable actions increase. However, this case-specific effect can only be treated by a full-probabilistic approach and thus is not further considered in this bulletin. In general the variable load effect depends on the time-variant component q(t) and on the time-invariant component C 0 (including model uncertainties). In most cases the maxima of the load effect related to t ref can be obtained as a product of both components: Q tref = C 0 max tref [q(t)] = C 0 q tref (4.2-8) Indicative probabilistic models for the time-invariant and time-variant components of selected variable loads are given in Table 4.2-1, JCSS (2006), Holický & Sýkora (2011) and Holický et al. (2009). More detailed data (e.g. for imposed loads in common types of buildings) are provided by JCSS (2006). The models in Table should be considered as informative and should always be carefully revised taking into account actual loading, structural conditions, and experimental data relevant for a particular structure. Table 4.2-1: Indicative probabilistic models of selected variable loads. X Variable Distr. µ X / X k δ X C 0W v b W Time-invariant component of the wind action including model uncertainties Annual maxima of the basic wind velocity Annual maxima of the basic wind pressure LN LN * Gum 0.65 ~ 1 / {1 - δ vb [ ln(-ln 0.98)]} ~ (1 + δ vb 2 ) / {1 - δ vb [ ln(-ln 0.98)]} ** *** C 0S S Time-invariant component of the snow load including model uncertainties Annual maxima of snow load on the ground LN Gum 1 ~ 1 / {1 - δ S [ ln(-ln 0.98)]} 0.15 ** C 0Q Q Model uncertainty in the load effect 5-year maxima of the imposed load LN Gum C 0T T Time-invariant component of the traffic load on road bridges including model uncertainties Annual maxima of traffic load on road bridges LN Gum * Three-parameter lognormal distribution. ** Should be based on meteorological data. Some indicative values of V S : (lowlands in the Czech Republic), (mountains in the Czech Republic). Value of V vb should always be based on local meteorological data since it is dependent on terrain roughness, orography and altitude. *** δ W δ vb (4 - δ vb 2 + 6δ vb ω vb ) 0.5 / (1 + δ vb 2 ) where ω denotes a sample skewness (in the case of insufficient data, Gumbel distribution may be assumed and ω vb = 1.14 may be considered). 5 Last modified:

6 Assuming the Gumbel distribution of the time-variant component, the mean of q tref is obtained as: µ q,tref = µ q σ q ln (t ref / t 0 ) (4.2-9) where t 0 is the basic reference period for q(t) (e.g. 1 year for climatic loads, 5 years for imposed loads). The standard deviation remains the same, i.e. σ q = σ q,tref. Note that other theoretical models such as a three-parameter lognormal distribution may be more suitable than the Gumbel distribution for some variable loads. In many cases it can be considered as an approximation that Q tref has a Gumbel distribution with the following parameters: µ Q,tref µ C0 µ q,tref ; δ Q,tref ( δ C0 2 + δ q,tref 2 + δ C0 2 δ q,tref 2 ) (4.2-10) where δ q,tref = σ q / µ q,tref. Consequently, the partial factor is assessed as: γ Q (µ Q,tref / Q k ) [1 - δ Q,tref ( ln(-ln Φ(-α E β t )))] (4.2-11) where Q k is the characteristic value applied in the assessment. Considering the probabilistic models given in Table 4.2-1, the partial factors γ Q obtained by (4.2-7) are shown in Figures to for various coefficients of variation δ q and different t ref, different β and α E = It is emphasised that the partial factors are significantly dependent on the assumed probabilistic models. In the reliability verification of a particular structure, probabilistic models should be carefully specified taking into account actual loading, structural conditions, and relevant experimental data. More detailed analysis should be based on a full-probabilistic approach. 6 Last modified:

7 fib SAG7 - group A.2 - draft of chapter 4.2 (Holicky & Sykora) a) b) Figure 4.2-3: Variation of the partial factor γ Q for the wind action with: a) the coefficient of variation of annual maxima of the basic wind velocity (t ref = 15 or 50 years), b) the reference period (δ vb = 0.15). a) b) Figure 4.2-4: Variation of the partial factor γ Q for the snow load with: a) the coefficient of variation of annual maxima of the snow on the ground (t ref = 15 or 50 years), b) the reference period (δ S = 0.6). 7 Last modified:

8 fib SAG7 - group A.2 - draft of chapter 4.2 (Holicky & Sykora) a) b) Figure 4.2-5: Variation of the partial factor γ Q for the imposed load with: a) the coefficient of variation of 5-year maxima (t ref = 15 or 50 years), b) the reference period (δ q = 1.1). a) b) Figure 4.2-6: Variation of the partial factor γ Q for the traffic load on road bridges with: a) the coefficient of variation of annual maxima (t ref = 15 or 50 years), b) the reference period (δ q = 0.03). 8 Last modified:

9 4.2.5 Examples The structural reliability of an existing reinforced concrete beam and short column exposed to a permanent and variable load is analysed by a full-probabilistic method. A single variable load represents the imposed load in a building (alt. 1) or snow load (alt. 2). Partial factors are obtained by the procedures described in section 4.2 for the target reliability index β t = 3.1. Measurements of material and geometry properties are available and statistical characteristics of the snow load on the ground are provided by a meteorological institute. Actions A characteristic value of the permanent action G k = µ G and its coefficient of variation δ G are estimated from measurements on the structure (see Table 4.2-2). G k = 50 knm (bending moment) and G k = 3.5 MN (axial compressive force) apply for the beam and short column, respectively. Assuming γ Sd,g = 1.0, the partial factor γ G is obtained from Figure 4.2-2; ξ = 0.85 is accepted from EN 1990 (2002). To cover a wide range of load combinations, the load ratio χ (study parameter) is introduced: Q G + Q k χ = (4.2-12) k k In general the load ratio may vary within the interval from nearly 0 (underground structures, foundations) up to nearly 1 (local effects on bridges, crane girders). For reinforced concrete beams in buildings it is expected to vary within the range from 0.4 up to 0.7; for columns within the range from 0.1 to 0.6. Given G k and χ, the characteristic value Q k can be obtained from Equation (4.2-12). Alt. 1: for the imposed load, it is assumed that no specific information on statistical characteristics of the load is available. The load effect Q k obtained from (4.2-12) is assumed to correspond to an effect of the imposed load in a relevant code such as EN (2002); the partial factors γ Q = 1.5 (independent of the reference period) and ψ 0 = 0.7 are accepted from EN 1990 (2002). Note that a more refined analysis can be based on the information provided by JCSS (2006). Based on the twin expressions (4.2-2a,b), the design load effect is obtained as follows: E d = max[γ G G k + γ Q ψ 0 Q k (χ); ξ γ G G k + γ Q Q k (χ)] (4.2-13) The statistical characteristics of the imposed load, considered in the reliability analysis, are provided in Table The imposed load is assumed to change each 5 years on average. The mean of the imposed load maxima related to the reference period t ref (= remaining working life in years; study parameter) is derived in case of a Gumbel distribution as follows: µ q,tref = µ q, σ q,5 ln (t ref / 5) (4.2-14) where σ q,5 = σ q,tref are standard deviations of maxima of the time-variant component of the imposed load related to 5 years and reference period, respectively. 9 Last modified:

10 Table 4.2-2: Probabilistic models for basic variables. X Variable, source of information Distr. Unit µ X / X k δ X γ X (β = 3.1) Other partial factors G Permanent action, in-site measurements N Figure ξ = 0.85 EN 1990 (2002) C 0Q Model uncertainty in the imposed load effect, expert judgement LN Q 5-year maxima of the imposed load, JCSS (2006), Holický & Sýkora (2011) Gum C 0S Time-invariant component of the snow load, JCSS (2006), Holický & Sýkora (2011) S Annual maxima of the snow load on the ground, meteorological data Gum EN 1990 (2002) ψ 0 = 0.7 EN 1990 (2002) LN for t ref = 15 y. Eq. (4.2-11) ψ 0 = 0.5 EN 1990 (2002) f c Concrete compressive strength, material tests LN MPa 30 / γ c = 1.13 Figure γ Rd1 γ Rd2 = section f y Yield strength of reinforcement, material tests LN MPa 500 / γ s = 1.05 Figure h Height of the beam (0.5 m), in-site measurements N m b Width of the beam (0.3 m), in-site measurements det m b col Width and height of the column (study parameter) N m m / x nom γ Rd1 γ Rd2 = section a Distance of reinforcement from surface, in-site measurements Gamma m K R Resistance uncertainty for the beam, Holický & Retief (2010) LN K R,col Resistance uncertainty for the column, expert judgement LN Last modified:

11 Alt. 2: for the snow load, statistical characteristics of annual maxima of the snow load on the ground are available (the Prague airport area, µ S,1 = 0.25 kn/m 2 and δ S,1 = 0.65). The characteristic value s k = 1 kn/m 2 is obtained from the Czech snow map. The load effect Q k obtained from (4.2-12) corresponds to the characteristic value s k multiplied by characteristic values of relevant factors (shape, exposure and thermal coefficients, loading width, span, etc.) C k (χ): C k (χ) = Q k (χ) / s k (4.2-15) According to section 4.2.4, γ Q is obtained from Equation (4.2-11) for t ref [years]. ψ 0 = 0.5 is accepted from EN 1990 (2002). The design load effect is obtained as: E d = max[γ G G k + γ Q (t ref ) ψ 0 Q k (χ); ξ γ G G k + γ Q (t ref ) Q k (χ)] (4.2-16) Statistical characteristics of the snow load considered in the probabilistic reliability analysis are provided in Table Resistance As a prerequisite, it is assumed that design resistances of the investigated beam and column are equal to the design load effects given by (4.2-13) for alt. 1 and (4.2-16) for alt. 2, R d = E d. The structural resistance of an existing concrete beam subjected to bending is given as: R(ρ) = K R ρ b(h - a) f y [h - a - 0.5ρ(h - a)f y / f c ] (4.2-17) The probabilistic models of the basic variables are given in Table To satisfy the condition R d = E d, the limiting value of the reinforcement ratio is: ρ ( χ, ) 2E ( ) ( ) d χ, tref 2 h a fck γ C f ck γ C t = ref 1 1 (4.2-18) f yk γ S b Note that for the expected range of χ 0.4,0.7 and t ref 1 y.,50 y., the reinforcement ratio varies between % and reinforcement properties are governing the failure. The resistance of an existing, centrically loaded short square column is given as: R(b col ) = K R,col (f c + ρ f y ) b col 2 (4.2-19) The probabilistic models of the basic variables are given in Table In this case, the constant reinforcement ratio ρ = 1 % is considered and the column width/height b col is derived to achieve R d = E d : b col ( χ, t ) ref ck C ( χ, t ) E γ + ρ f d ref = (4.2-20) f yk γ S For the expected range of χ 0.1,0.6 and t ref 1 y.,50 y., b col varies between m. In the case of the column, the concrete failure mode is dominant. Reliability analysis The structural reliability of the reinforced concrete members is analysed by FORM. The limit state functions for the imposed and snow load are given by Eqs. (4.2-21a) and (4.2-21b), respectively: 11 Last modified:

12 Z(X) = K R(col) R(χ,t ref ) - G - C 0Q Q tr (χ) Z(X) = K R(col) R(χ,t ref ) - G - C k (χ) C 0S S (4.2-21a) (4.2-21b) Figure shows the variation of the reliability index β with the load ratio χ for t ref = 15 y. and β t = 3.1. To indicate the benefits of applying the design value method, the reliability index for an alternative set of the codified partial factors for new structures (i.e. γ C = 1.5, γ S = 1.15, γ G = 1.35, γ Q = 1.5), independent of χ and t ref, is plotted for the beam exposed to the snow load. It follows from Figure that: In the case of snow load the design value method captures well random properties of the basic variables and the obtained β values are reasonably close to the target level. In the case of column rather lower reliability levels for small χ ratios can be attributed to the underestimated factors γ Rd that should be in this case slightly increased (by about 5 %). For the imposed load, higher reliabilities are obtained since acceptance of γ Q = 1.5 seems to be rather conservative for t ref = 15 y. Influences of the target reliability (β t = 3.1 here) and of information obtained from the measurements are not adequately covered when the alternative set of the partial factors is considered (particularly γ G can be reduced when measurements indicate low variability of G). Thus, relatively high reliability levels are obtained. Figure indicates variation of the reliability index β with the reference period t ref for the beam exposed to the snow load and χ = 0.5. The design value method captures well influence of t ref while the reliability is considerably decreasing with an increasing t ref when the alternative set of partial factors is considered. Concluding remarks for the examples The following conclusions may be drawn from the presented numerical examples: The structural reliability of reinforced concrete members verified by the design value approach is close to specified target reliability for any t ref and χ. The effect of a reduced uncertainty due to measurements can be easily included. 12 Last modified:

13 fib SAG7 - group A.2 - draft of chapter 4.2 (Holicky & Sykora) (a) Variation of β with χ for the imposed load the beam exposed to (b) Variation of β with χ for the beam exposed to the snow load (c) Variation of β with χ for to the imposed load the column exposed (d) Variation of β with χ for the column exposed to the snow load Figure 4.2-7: Variation of the reliability index β with the load ratio χ for t ref = 15 y. and β t = Last modified:

14 fib SAG7 - group A.2 - draft of chapter 4.2 (Holicky & Sykora) Figure 4.2-8: Variation of the reliability index β with the reference period t ref for the beam exposed to the snow load and χ = Literature EN 1990: Eurocode - Basis of structural design. Brussels: CEN. EN : Eurocode 1: Actions on structures - Part 1-1: General actions; Densities, self-weight, imposed loads for buildings. Brussels: CEN. EN : Design of concrete structures - Part 1-1: General rules and rules for buildings. Brussels: CEN. Holický, M. & Retief, J.V Probabilistic basis of present codes of practice. In A. Zingoni (ed.), Proc. SEMC 2010, University of Cape Town, 6-8 September Taylor & Francalibration of codes. cis. Holický, M. & Sýkora, M Conventional probabilistic models for In M.H. Faber, J. Köhler & K. Nishijima (eds.), Proc. ICASP11, ETH Zurich, 1-4 August Leiden: CRC Press/Balkema. Holický, M., Marková, J. & Sýkora, M Assessment of Deteriorating Reinforced Concrete Road Bridges. In R. Bris, C. Guedes Soares & S. Martorell (eds.), Proc. ESREL 2009, Prague, Czech Republic, 7-10 September Leiden: CRC Press/Balkema. ISO 13822: Bases for design of structures - Assessment of existing structures. Geneve, Switzerland: ISO TC98/SC2. ISO 2394: General principles on reliability for structures. Geneve, Switzerland: ISO. JCSS JCSS Probabilistic Model Code. Zurich: Joint Committee on Structural Safety. < 14 Last modified:

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16 Working notes on further development It should be clarified somewhere in the bulletin how to address the following: 1) statistical uncertainty related to small samples influence on CoV (material properties, permanent actions) 2) degradation aspects 3) which measurements should be taken to assess concrete strength 4) the partial factor γ m and the characteristic value shall be derived from the actual distribution of the material property under consideration (based on prior information, or results of tests or the combination of both) Other comments: 5) Load combinations (subsection 4.2.1) and splitting of the partial factors (e.g. model uncertainty * geometry * material Eqs. (4.2-3) and (4.2-5)) could be incorporated in section 4.1 since it is likely to be the same for sections 4.2 and Last modified:

17 fib SAG7 - group A.2 - draft of chapter 4.2 (Holicky & Sykora) Background information - comparison of estimation of γ Q using a full-probabilistic approach (4.2-7) and approximation using the Gumbel distribution (4.2-11). Compared to the 1 st draft, in the 2 nd draft the alternative based on two separate partial factors for time- It is foreseen that invariant and time-variant component of a variable action is not included. the approximation based on Gumbel distribution is sufficiently simple and yet accurate for practical applications. Numerical experience indicates that differences between (4.2-7) and (4.2-11) are acceptable for typical characteristics of wind, imposed and traffic loads. To illustrate this, the following input data for the wind action are considered: - C 0W ~ LN(µ X / X k = 0.65; δ X = 0.3) - annual maxima of the basic ln 0.98)]}; δ X = 0.1). wind velocity v b (Gumbel, µ X / X k = 1 / {1 - δ X [ ln(- Partial factors γ Q for different t ref (β = 3.8, and α E = -0.7) are shown in the following figure. Variation of the partial factor γ Q for the wind action with t ref. For the snow load the following input data are considered: - C 0S ~ LN(µ X / X k = 1; δ X = 0..15) - annual maxima S based on meteorological data for: a) lowlands in the Czech Republic, typically with a higher CoV than for mountains (Gumbel, µ X / X k = 1 / {1 - δ S [ ln(-ln 0.98)]}; δ S = 0.65) b) highlands in the Czech Republic (Gumbel, µ X / X k = 1 / {1 - δ S [ ln(-ln 0.98)]}; δ S = 0.5). Partial factors γ Q for different t ref (β = 3.8, and α E = -0.7) are shown in the following figure. 17 Last modified:

18 fib SAG7 - group A.2 - draft of chapter 4.2 (Holicky & Sykora) a) Variation of the partial factor γ Q a) lowlands, b) highlands. for the snow load with t ref for: b) 18 Last modified: