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Reliability assessment of vertically loaded masonry walls
Structural Safety 62 (2016) 47–56
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Structural Safety journal homepage: www.elsevier.com/locate/strusafe
Reliability assessment of vertically loaded masonry walls Tammam Bakeer Technische Universität Dresden, Germany
a r t i c l e
i n f o
Article history: Received 4 November 2015 Received in revised form 2 June 2016 Accepted 3 June 2016 Available online 10 June 2016 Keywords: Partial safety method Full probabilistic method Limit state function First order reliability method Estimating the coefficient of variation ECOV Monte Carlo simulation MCS Bivariate normal distribution Latin hypercube sampling method LHS Hasofer–Lind transformation
a b s t r a c t For the safety assessment of vertically loaded masonry walls, the Eurocode 6 considers a linear limit state function with only one basic parameter for the compressive strength of masonry. This assumption disregards the contribution of the uncertainty in other material parameters, like the elastic modulus, which is more important than the compressive strength in slender masonry walls. Furthermore, Eurocode 0 gives no clear advice on how to deal with the partial safety factors for non-linear problems in general. A reliability assessment has been carried out for the vertically loaded masonry walls, in order to clearly understand the contribution of the uncertainty of different material parameters on the evaluation of safety. The limit state function of the problem has been calculated using the transfer-matrix method. The full probabilistic approach has been applied using Monte Carlo simulations which helped to propose a new approach for the use of the partial safety factors in the non-linear analysis. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In recent decades, a great progress has been achieved in computational techniques and engineering software, which made the non-linear structural analysis possible for the practising engineer. Many codes of practise today are based on the partial safety factors which can be used for problems defined by linear limit state function. However, the code provisions give no advice about how to deal with the partial safety factors in the non-linear analysis. The provisions given in EN 1990 are rather general and leave the decision to the practising engineer. In some codes, a new safety format on the basis of the global safety factors was introduced, e.g. DIN 1045-1 and EN 1992-2. In EN 1996-1-1, the limit state function defining the problem of vertically loaded masonry walls has been considered linear with one basic material parameter for the compressive strength of masonry. One partial safety factor cM has been defined for the material which takes into account the uncertainty in the compressive strength. However, for slender masonry walls, where stability failure occurs due to the second order effect, the vertical load bearing capacity becomes dependent from the elastic modulus but not from the compressive strength. This nonlinear behaviour leads to the question of the basic approach of the partial safety factors and the related material properties in numerical analysis of masonry structures. E-mail address: [email protected] http://dx.doi.org/10.1016/j.strusafe.2016.06.004 0167-4730/Ó 2016 Elsevier Ltd. All rights reserved.
In the following sections, the available safety formats are going to be reviewed and the resistance of the wall will be modelled in both global and partial safety formats.
2. Safety formats Several proposals for the appropriate safety format for nonlinear analysis of concrete structure were proposed (see e.g. Schlune et al. [1,2], Cervenka [3], fib model code 2010 [4], and Allaix [5]). But the current safety formats doesn’t properly account for modelling the uncertainty in non-linear analysis. In the following, the safety formats based on partial factors, global factors, and probabilistic analysis are going to be discussed. 2.1. Partial factor format The European standards use reliability verification approach on the basis of partial factor method. It is an appropriate format for linear limit state functions. This format describes the uncertainties in the variables by means of the design values assigned to each variable. The design values of the variables are usually introduced in terms of their representative values or characteristic values. The design resistance can be obtained as following:
Rd ¼
Rðf d Þ
cRd
ð1Þ
48
T. Bakeer / Structural Safety 62 (2016) 47–56
where cRd is the safety factor considering the uncertainty in structural resistance, fd is the design value of material strength and is given by:
fd ¼ g
fk
ð2Þ
cm
where fk is the characteristic strength of the material, g is a conversion factor to correct the resistance for durability effect, cm is a partial safety factor considers the uncertainty in material properties. The following simplification of expression (1) can be made:
f Rd ¼ R g k
ð3Þ
cM
f k ¼ lf ekn V f
ð9Þ
The partial safety factor can be determined for lognormal distribution as following:
cf ¼ eðkn þbaf ÞV f
ð10Þ
By taking the coefficient of variation of material strength Vf = 0.194 with target reliability index on annual basis b = 4.7 and sensitivity factor af = 0.8, gives cf = cM = 1.5. 2.2. Global resistance factor format
where
cM ¼ cm cRd
ð4Þ
The design value for single action can be obtained as following:
Sd ¼ SðcF F rep Þ
ð5Þ
where cF is the partial safety factor of action. Frep is the representative value of action. Considering a normal distribution for the strength of masonry with mean value lf and standard deviation rf, the characteristic strength can be obtained by the following equation (Fig. 1):
f k ¼ lf ð1 kn V f Þ
ð6Þ
kn ¼ 1:645 when the number of tests goes to infinity. If the number of tests is known, kn can be determined from table D1 in EN 1990. The partial safety factor for the material strength cf can be obtained as a ratio of the characteristic value to the design value and given by:
cf ¼
The above equation can be reduced to the following simple expression:
1 kn V f fk ¼ f d 1 b af V f
ð7Þ
By taking the coefficient of variation of material strength Vf = 0.125 with a target reliability index on nnual basis b = 4.7 (EN 1990 [6] for reliability class RC2 and 1 year reference period) and sensitivity factor af = 0.8, gives cf = 1.5. If the distribution of the material strength is lognormal, the characteristic values can be given as following:
lf
pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2
kn
f k ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e V 2f þ 1
lnðV f þ1Þ
ð8Þ
The global resistance format introduced to consider the nonlinearity of the limit state function which cannot be considered in the partial safety format [4]. The calculation of design resistance based on the mean value takes the following form:
Rd ¼
lR c~R cRd
ð11Þ
cRd is the model uncertainty factor and takes into account the quality of the resistance model. CEB-FIB code [4] recommends cRd = 1.0 for no uncertainties, cRd = 1.06 for low uncertainties, and cRd = 1.1 for high uncertainties. c~R global resistance factor related to the mean value and can be estimated from full probabilistic analysis. However, for special cases when the distribution of the resistance is known, the global resistance safety factor can be calculated by the method of estimating the coefficient of variation ECOV. In ECOV, the coefficient of variation can be estimated from two non-linear analysis at the mean and the characteristic values of the material strength. If the resistance is normally distributed, the coefficient of variation VR can be calculated as following: VR ¼
1 Rk 1 1:65 lR
ð12Þ
The global central resistance factor can be calculated as follows
c~R ¼
1 1 b aR V R
ð13Þ
If the resistance is log-normally distributed, the coefficient of variation VR can be calculated as following:
Fig. 1. Determination of the characteristic value of the material parameter in case of normal distribution.
T. Bakeer / Structural Safety 62 (2016) 47–56
VR ¼
1 l ln R 1:65 Rk
49
ð14Þ
The global central resistance factor can be calculated as follows
c~R ¼ ebaR V R
ð15Þ
2.3. Full probabilistic approach The design value of resistance is estimated from the probabilistic distribution of the resistance as following:
PðR Rd 6 0Þ ¼ F n ðaR bÞ
ð16Þ
If CDF is the cumulative distribution function of the resistance R, then the design value of resistance is given by:
Rd ¼ CDF 1 fF n ðaR bÞg
ð17Þ
The probabilistic approach can be applied by running a virtual experiment with a large number of random sample data. This procedure is known as Monte Carlo simulation MCS and requires to generate random plausible sampling data of the variables influencing the resistance e.g. material properties. An array of resistance values is generated based on the sampling data, which represent the distribution of the global resistance. The mean, the standard deviation, and the coefficient of variation can be calculated. This fully describes the random properties of resistance. The design value of resistance can be estimated using the reliability index. 3. Nonlinear limit state function
Fig. 2. Cross-section of vertically loaded masonry wall, showing the stress and deformation state.
Since masonry is a material with low tensile strength, the wall may crack under certain conditions leading to further complications due to the reduction in the effective cross-section. Masonry members under compression may fail either because of material over-stressing for squat members or because of stability failure for slender members. For squat masonry members, the failure takes place if the compressive strain at any cross-section reached the ultimate compressive strain of the material. Nevertheless, for slender masonry elements the failure occurs before reaching the ultimate compressive strain of the material at any cross-section. The former mode of failure called material failure and the later called stability failure. Both failure modes are going to play crucial roles in the determination of the capacity reduction factor. For masonry wall with a height h and a thickness t under vertical eccentric loading (Fig. 2), the load bearing capacity is mainly influenced by the eccentricity of the load eo, and the slenderness of the wall k. The slenderness ratio k is defined as following:
k¼
h t
rffiffiffi f E
ð18Þ
where f is the compressive strength of masonry, and E is the initial elastic modulus. The resistance of the wall is defined by the following expression:
NR ¼ f U t
ð19Þ
where NR the resistance load bearing capacity [force/length]. This form has been adopted in many standards to calculate the load bearing capacity. However, more convenient form for the resistance can be defined as following:
nR ¼ f U
ð20Þ
where nR is the strength capacity of the wall due to slenderness and eccentricity. The above equation is useful to demonstrate the influence of the second order effect and eccentricity on the load bearing
capacity by the means of the capacity reduction factor U. A detailed description on determination of the factor U can be found in [7–9]. For short walls, the maximum possible value of nR can be obtained when U = 1. The compressive strength in Eq. (20) can be written with respect to the elastic modulus as following:
f ¼E
k2 r 2h
ð21Þ
where rh = h/t. By substituting in Eq. (20), another demonstration of the stress capacity in term of elastic modulus can be obtained:
nR ¼ E
k2 U r 2h
ð22Þ
For slender walls the maximum possible value of nR can be obtained when:
k2 U ¼
p2 12
ð23Þ
The above value corresponds to the Euler’s buckling load:
nR ¼
E p2 r 2h 12
ð24Þ
The capacity reduction factor has been calculated numerically using the transfer-matrix method described in [10,8]. A graphical representation of the relation between k2 U and k2 at different eccentricities are plotted in Fig. 3. This shows that k2 U reaches a constant value at stability failure. The basic variables of the limit state function for nR are the compressive strength and the elastic modulus:
nR ¼ nR ðf ; EÞ
ð25Þ
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T. Bakeer / Structural Safety 62 (2016) 47–56
Fig. 3. Graphical representation of the relationship between k2 and U k2 showing the areas for material failure and stability failure.
If there is no eccentricity, the limit state function takes the following stepwise form:
nR ðf ; EÞ ¼
8 <
k < 2pp ffiffi3
f;
: rE2 p12 ; 2
h
k P 2pp ffiffi3
ð26Þ
The graphical representation of the limit state function is presented in Figs. 4 and 5 for an example with h/t = 20 and e0/ t = 0.2. It can be noted, from Fig. 5, the stability failure is dominant for bigger values of compressive strength, while the material failure is dominant for bigger values of the elastic modulus. The contours of equal resistances get parallel to f axis at the domain of stability failure. This indicates the independency of the resistance from the compressive strength of masonry at this domain. However, in the domain of material failure, the contours of equal resistances get parallel to E axis only for large values of E. The resistance in this domain is dependent from both E and f. 4. Resistance modelling In many design codes of masonry, the resistance has been modelled considering the uncertainty in the compressive strength of the material. In Eurocode EN 1996-1-1 [12], the resistance of unreinforced masonry subjected to mainly compression action is given by:
NRd ¼ U t f d
ð27Þ
fd is the design compressive strength. By substituting the value of fd from Eq. (2) in Eq. (27), leaves:
NRd ¼ U t g
fk
cm
¼
1
cm
Nðf k Þ
ð28Þ
where N(fk) is a function of the characteristic compressive strength of masonry fk. This shows that the partial safety factor of the material cm has been introduced as a material factor but becomes equal to the global resistance factor as the randomness in U is neglected. Nevertheless, the modelling of the resistance cannot be realistic without considering the uncertainty in the elastic modulus as well. For walls with small slenderness ratio the uncertainty in compressive strength is important but the uncertainty in elastic modulus is not important, on another hand for walls with big slenderness ratio the uncertainty in elastic modulus becomes important for the design while the uncertainty in compressive strength can be neglected. In the following, an example is given considering the uncertainty in the compressive strength but not in elastic modulus. The global safety factor of resistance cR has been calculated using Monte Carlo simulation with one variable for different ratios of h/t. The distribution histograms of the resistance have been demonstrated in Fig. 6. The results show that the partial safety factor cR becomes less for higher slenderness because the influence of the elastic modulus on the resistance becomes less. This shows the importance of using two basic variables (i.e. the compressive strength and elastic modulus) to model the uncertainty in resistance. In the Eurocode 2 (EN 1992-1-1 [13], Section 5.8.6), a partial safety factor has been introduced for elastic modulus to be considered in the design of slender concrete elements.
T. Bakeer / Structural Safety 62 (2016) 47–56
51
Fig. 4. 3D representation of the stress capacity surface in terms of compressive strength and elastic modulus for wall with h/t = 20 and e0/t = 0.2.
Fig. 5. The limit state function of the stress capacity in terms of compressive strength and elastic modulus for wall with h/t = 20 and e0/t = 0.2. The red area represents the practical domain at which the material failure occurs while the blue area represents the practical domain at which the stability failure occurs. The practical domain is considered for compressive strengths f = 2–16 MPa and E/f = 150–2000. In Germany, the practical range of E/f is between 500 for masonry with autoclaved aerated concrete bricks and 2700 for masonry with concrete bricks, (see Table 12 of DIN EN 1996-1-1 NA [11]). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
52
T. Bakeer / Structural Safety 62 (2016) 47–56
Fig. 6. Histograms for resistance distribution at e0/t = 0.15 generated for a mean compressive strength lf = 5 MPa and considering no uncertainty in the elastic modulus E = 3500 MPa. Two distributions of the compressive strength are considered, (a) normal distribution, and (b) lognormal distribution. The global resistance safety factor is determined by Monte Carlo simulation.
In the following sections, the resistance of masonry has been modelled using two basic variables and the safety factors have been determined for the global safety format and for the partial safety format. 4.1. Global safety format In the global safety format of resistance, the design resistance is given in the following format:
NRd ¼
1
cR
NR ðf k ; Ek Þ
ð29Þ
The joint distribution of the compressive strength f and the elastic modulus E is considered as bivariate normal distribution with no correlation between f and E. The design value of the resistance can be determined by Eq. (16). Considering the nonlinear limit state function of resistance defined in Section 3, the design value of resistance can be estimated by Monte Carlo simulation. The procedure to estimate the global safety factor of resistance can be summarized in the following steps: Step 1: Generation of sampling data for f and E is performed using Latin hypercube sampling method LHS, also known as
T. Bakeer / Structural Safety 62 (2016) 47–56
stratified sampling technique. It represents multivariate sampling method that guarantee non-overlapping designs (Choi et al. [14] and Stein [15]). Step 2: Using the generated sampling data, the resistance can be calculated at each pair of f, E based on the defined limit state function. This gives the probability distribution of resistance. Step 3: Calculation of the design resistance at probability of failure equal to F n ðaR bÞ:
nRd ¼ CDF 1 fF n ðaR bÞg
ð30Þ
Step 4: The global safety factor of resistance can be determined by:
cR ¼
nR ðf k ; Ek Þ nRd
ð31Þ
53
The function nR is demonstrated for an example with e0/t = 0.2 and h/t = 20 together with sampling points generated by LHS method in Fig. 7. The following data were used to generate the sampling points: mean values lf = 5 MPa and lE = 3.5 103 MPa with coefficient of variations Vf = VE = 0.125. The design value of resistance nRd = 1.2 MPa is determined at probability of failure
equal to F n ðaR bÞ ¼ 8:5 105 . The calculated resistance at the characteristic values is nR(fk, Ek) = 1.61 MPa. Consequently, the global safety factor of resistance for this example is cR = 1.35. The same procedure to calculate the global safety factor of resistance is repeated for different values of slenderness ratio and eccentricity. The results are plotted in Fig. 8. It can be noted from Fig. 8 that cR takes the value of 1.5 at extreme values of slenderness i.e. for big and small slenderness ratios. However, cR can reach values less than 1.3 for slenderness k ¼ 0:9 and relative eccentricity e0/t = 0.1. The results were calcu-
Fig. 7. An example showing the determination of the design resistance using sampling data generated by LHS method. The area at which the probability of failure Pf is less than F n ðaR bÞ ¼ 8:5 105 is highlighted in blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. The global safety factor of resistance cR based on the slenderness k and relative eccentricity e0/t estimated by Monte Carlo simulation. The compressive strength and elastic modulus assumed to have equal coefficients of variation Vf = VE = 0.125.
54
T. Bakeer / Structural Safety 62 (2016) 47–56
lated assuming equal coefficient of variations for compressive strength and elastic modulus, but if the coefficient of variation for elastic modulus is more than 0.125 it is expected to get values of cR more than 1.5 for big slenderness ratios. 4.2. Partial safety format In the partial safety format, the design resistance is given in the following format:
NRd ¼ NR ðf d ; Ed Þ
ð32Þ
where the design values of the compressive strength and elastic modulus are given as following:
fd ¼
fk
cf
¼
lf Ek l and Ed ¼ ¼ E c~f cE c~E
ð33Þ
cf and cE are the partial safety factors of the compressive strength and elastic modulus respectively with respect to the char~E are the partial safety factors of the com~f and c acteristic values. c pressive strength and elastic modulus, respectively, with respect to the mean values. The partial safety factors can be determined approximately according to the first-order second-moment theory (Melchers [16]) using Hasofer–Lind transformation [17]. The procedure can be described in the following steps: Step 1: Assume initial values of af and aE e.g. af = 1 and aE = 0; Step 2: Calculate the design point on the basis of the assumed a values:
f d ¼ lf ð1 af V f aR bÞ
ð34Þ
Ed ¼ lE ð1 aE V E aR bÞ
ð35Þ
Step 3: Calculation of the partial derivatives of the limit state function at the assumed design point. This can be performed numerically by assuming small values for each of Df and DE in the following equations:
@nR nR ðf d þ Df ; Ed Þ nR ðf d ; Ed Þ ðf ; Ed Þ ¼ @f d Df
ð36Þ
@nR nR ðf d ; Ed þ DEÞ nR ðf d ; Ed Þ ðf ; Ed Þ ¼ @E d DE
ð37Þ
Step 4: Calculation of the sensitivity factors af and aE:
@nR rf @f af ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 @nR R rf þ @n rE @f @E
aE ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a2f
ð38Þ
ð39Þ
Step 5: Checking the newly calculated values of af and aE with the old ones. If the difference is big, the new calculated values should be used again in Step 2; Step 6: Calculation of the partial safety factors with respect to the characteristic values for the compressive strength and elastic modulus:
cf ¼
1 kn V f 1 af V f aR b
ð40Þ
cE ¼
1 kn V E 1 aE V E aR b
ð41Þ
Using the above procedure the partial safety factors has been calculated for an example with h/t = 20 e0/t = 0.3 considering different slenderness ratios. The results are demonstrated in Fig. 9. The results can be explained based on the type of failure. In the area of compression failure, the partial safety factor of compressive strength cf = 1.5 and for elastic modulus cE = 0.8. In the area of stability failure, the values turn to cE = 1.5 and cf = 0.8. For the illustration of these results, three points for the mean values M1, M2, and M3 are considered in Hasofer–Lind transformation space (Fig. 10). The first point M1 chosen in space to get no influence of the elastic modulus on the resistance, while the third point M3 chosen to get no influence of the compressive strength. The second point M2 represent the general case where both elastic modulus and compressive strength contribute to the resistance. At point M1 the design value of the elastic modulus is equal to the mean value Ed1 = lE1. This gives af = 1 and aE = 0. Considering equal coefficient of variation for both compressive strength and elastic modulus Vf = VE = 0.125, the partial safety factors can be obtained directly by applying Eqs. (40) and (41):
Fig. 9. Partial safety factors for the compressive strength and elastic modulus calculated for an example with h/t = 20 and e0/t = 0.3.
T. Bakeer / Structural Safety 62 (2016) 47–56
55
Fig. 10. Geometric interpretation of the possible design points in Hasofer–Lind space.
cf ¼
1 1:65 V f ¼ 1:5 1 0:125 0:8 4:7
cE ¼ 1 1:65 0:125 ¼ 0:8
ð42Þ ð43Þ
At point M3 the design value of the compressive strength is equal to the mean value fd3 = lf3. This gives af = 0 and aE = 1. By applying Eqs. (40) and (41) gives cf = 0.8 and cE = 1.5. Point M2 represents the range of k in Fig. 9 from 0.3 to 0.79. The partial safety factors can be calculated by iteration. 5. Recommendations for non-linear analysis Eurocode EN 1990 gives no regulations on how to deal with safety factors in case of non-linear analysis. Nevertheless, based on the investigation carried out in Section 4.1 on the global resistance safety factor, if the same safety factors of linear limit state functions applied for the non-linear one, it will result in conservative design. Therefore, it is recommended according to Section 4.2 to apply partial safety factors for the compressive strength and elastic modulus, so that the design resistance can be determined as following:
NRd ¼ min½NR ðf d ; lE Þ; NR ðlf ; Ed Þ
ð44Þ
Since the failure state of the structure element is unknown before analysis, it is required to run two non-linear analysis one using the design value of the compressive strength together with the mean value of the elastic modulus, and the other one using the mean value of the compressive strength together with the design value of the elastic modulus. The analysis which gives the minimum capacity is decisive. 6. Concluding remarks In the code of practise of masonry EN 1996-1-1, the resistance of vertically loaded masonry wall has been modelled considering
the uncertainty in the compressive strength of the material. However, for slender masonry walls the uncertainty in the elastic modulus becomes more important than the uncertainty in the compressive strength. The present study suggests modelling the resistance considering the uncertainty in both material parameters, the compressive strength, and the elastic modulus. This is why in EN 1996-1-1, the capacity reduction factor for eccentricity and slenderness U cannot be considered as a constant parameter in the resistance model, but it must be considered as a function of two basic parameters. For a wall with small slenderness ratio, the uncertainty in compressive strength is more important than the uncertainty in elastic modulus. On another hand, for a wall with big slenderness ratio the uncertainty in elastic modulus becomes more important for the design. The partial safety factors are variable and depend mainly on the slenderness ratio of the wall. However, for practical use of partial safety factors, it is suggested to run two non-linear analysis one using the design value of the compressive strength together with the mean value of the elastic modulus, and the other one using the mean value of the compressive strength together with the design value of the elastic modulus. The analysis which gives the minimum capacity is the crucial one.
References [1] Schlune H, Plos M, Gylltoft K. Safety formats for nonlinear analysis tested on concrete beams subjected to shear forces and bending moments. Eng Struct 2011;33(8):2350–6. [2] Schlune H, Plos M, Gylltoft K. Safety formats for non-linear analysis of concrete structures. Mag Concr Res 2012;64(7):563–74. [3] Cervenka V. Global safety format for nonlinear calculation of reinforced concrete. Beton- und Stahlbetonbau 2008;103(S1):37–42. [4] CEB-FIP: Model Code. First complete draft, 1. Lausanne, Switzerland: Internationnal Federation for Structural Concrete; 2010. [5] Allaix DL, Carbone VI, Mancini G. Global safety format for non-linear analysis of reinforced concrete structures. Struct Concr 2013;14(1):29–42.
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T. Bakeer / Structural Safety 62 (2016) 47–56
[6] EN 1990:2005-12+A1. Eurocode-basis of structural design. European Committee for Standardization; 2005. [7] Bakeer T. Empirical estimation of the load bearing capacity of masonry walls under buckling – critical remarks and a new proposal for the Eurocode 6. Constr Build Mater 2016;113:376–94. [8] Bakeer T. Stability of masonry walls Habilitation thesis. Technische Universität Dresden: Dresden; 2016. [9] Bakeer T, Jäger W, Pflücke T, Christiansen PD. Buckling of masonry with low modulus of elasticity. In: Lourenço Paulo B, Haseltine Barry, Vasconcelos Graça, editors. Proc. of 9th International Masonry Conference. Guimarães, Portugal: Department of Civil Engineering, University of Minho; 2014. paper 1235. [10] Bakeer T. Assessment the stability of masonry walls by the transfer-matrix method. Eng Struct 2016;110:1–20. [11] DIN EN 1996-1-1/NA: 2012–05: Nationaler Anhang – National festgelegte Parameter – Eurocode 6: Bemessung und Konstruktion von
[12]
[13]
[14] [15] [16] [17]
Mauerwerksbauten– Teil 1–1: Allgemeine Regeln für bewehrtes und unbewehrtes Mauerwerk. Deutsches Institut für Normung: Berlin Mai, 2012. EN 1996-1-1:2005-11. Eurocode 6: design of masonry structures – part 1-1: general rules for reinforced and unreinforced masonry structures. European Committee for Standardization; 2005. EN 1992-1-1:2004-12. Eurocode 2: design of concrete structures – part 1-1: general rules and rules for buildings. European Committee for Standardization; 2004. Choi S-K, Grandhi RV, Canfield RA. Reliability-based structural design. London: Springer-Verlag; 2010. Stein M. Large sample properties of simulations using latin hypercube sampling. Technometrics 1987;29(2):143–51. Melchers RE. Structural reliability analysis and prediction. 2nd ed. Chichester, England: John Wiley & Sons; 1999. Hasofer AM, Lind NC. Exact and invariant second-moment code format. J Eng Mech Div 1974;100(1):111–21.
Contents lists available at ScienceDirect
Structural Safety journal homepage: www.elsevier.com/locate/strusafe
Reliability assessment of vertically loaded masonry walls Tammam Bakeer Technische Universität Dresden, Germany
a r t i c l e
i n f o
Article history: Received 4 November 2015 Received in revised form 2 June 2016 Accepted 3 June 2016 Available online 10 June 2016 Keywords: Partial safety method Full probabilistic method Limit state function First order reliability method Estimating the coefficient of variation ECOV Monte Carlo simulation MCS Bivariate normal distribution Latin hypercube sampling method LHS Hasofer–Lind transformation
a b s t r a c t For the safety assessment of vertically loaded masonry walls, the Eurocode 6 considers a linear limit state function with only one basic parameter for the compressive strength of masonry. This assumption disregards the contribution of the uncertainty in other material parameters, like the elastic modulus, which is more important than the compressive strength in slender masonry walls. Furthermore, Eurocode 0 gives no clear advice on how to deal with the partial safety factors for non-linear problems in general. A reliability assessment has been carried out for the vertically loaded masonry walls, in order to clearly understand the contribution of the uncertainty of different material parameters on the evaluation of safety. The limit state function of the problem has been calculated using the transfer-matrix method. The full probabilistic approach has been applied using Monte Carlo simulations which helped to propose a new approach for the use of the partial safety factors in the non-linear analysis. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In recent decades, a great progress has been achieved in computational techniques and engineering software, which made the non-linear structural analysis possible for the practising engineer. Many codes of practise today are based on the partial safety factors which can be used for problems defined by linear limit state function. However, the code provisions give no advice about how to deal with the partial safety factors in the non-linear analysis. The provisions given in EN 1990 are rather general and leave the decision to the practising engineer. In some codes, a new safety format on the basis of the global safety factors was introduced, e.g. DIN 1045-1 and EN 1992-2. In EN 1996-1-1, the limit state function defining the problem of vertically loaded masonry walls has been considered linear with one basic material parameter for the compressive strength of masonry. One partial safety factor cM has been defined for the material which takes into account the uncertainty in the compressive strength. However, for slender masonry walls, where stability failure occurs due to the second order effect, the vertical load bearing capacity becomes dependent from the elastic modulus but not from the compressive strength. This nonlinear behaviour leads to the question of the basic approach of the partial safety factors and the related material properties in numerical analysis of masonry structures. E-mail address: [email protected] http://dx.doi.org/10.1016/j.strusafe.2016.06.004 0167-4730/Ó 2016 Elsevier Ltd. All rights reserved.
In the following sections, the available safety formats are going to be reviewed and the resistance of the wall will be modelled in both global and partial safety formats.
2. Safety formats Several proposals for the appropriate safety format for nonlinear analysis of concrete structure were proposed (see e.g. Schlune et al. [1,2], Cervenka [3], fib model code 2010 [4], and Allaix [5]). But the current safety formats doesn’t properly account for modelling the uncertainty in non-linear analysis. In the following, the safety formats based on partial factors, global factors, and probabilistic analysis are going to be discussed. 2.1. Partial factor format The European standards use reliability verification approach on the basis of partial factor method. It is an appropriate format for linear limit state functions. This format describes the uncertainties in the variables by means of the design values assigned to each variable. The design values of the variables are usually introduced in terms of their representative values or characteristic values. The design resistance can be obtained as following:
Rd ¼
Rðf d Þ
cRd
ð1Þ
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T. Bakeer / Structural Safety 62 (2016) 47–56
where cRd is the safety factor considering the uncertainty in structural resistance, fd is the design value of material strength and is given by:
fd ¼ g
fk
ð2Þ
cm
where fk is the characteristic strength of the material, g is a conversion factor to correct the resistance for durability effect, cm is a partial safety factor considers the uncertainty in material properties. The following simplification of expression (1) can be made:
f Rd ¼ R g k
ð3Þ
cM
f k ¼ lf ekn V f
ð9Þ
The partial safety factor can be determined for lognormal distribution as following:
cf ¼ eðkn þbaf ÞV f
ð10Þ
By taking the coefficient of variation of material strength Vf = 0.194 with target reliability index on annual basis b = 4.7 and sensitivity factor af = 0.8, gives cf = cM = 1.5. 2.2. Global resistance factor format
where
cM ¼ cm cRd
ð4Þ
The design value for single action can be obtained as following:
Sd ¼ SðcF F rep Þ
ð5Þ
where cF is the partial safety factor of action. Frep is the representative value of action. Considering a normal distribution for the strength of masonry with mean value lf and standard deviation rf, the characteristic strength can be obtained by the following equation (Fig. 1):
f k ¼ lf ð1 kn V f Þ
ð6Þ
kn ¼ 1:645 when the number of tests goes to infinity. If the number of tests is known, kn can be determined from table D1 in EN 1990. The partial safety factor for the material strength cf can be obtained as a ratio of the characteristic value to the design value and given by:
cf ¼
The above equation can be reduced to the following simple expression:
1 kn V f fk ¼ f d 1 b af V f
ð7Þ
By taking the coefficient of variation of material strength Vf = 0.125 with a target reliability index on nnual basis b = 4.7 (EN 1990 [6] for reliability class RC2 and 1 year reference period) and sensitivity factor af = 0.8, gives cf = 1.5. If the distribution of the material strength is lognormal, the characteristic values can be given as following:
lf
pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2
kn
f k ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e V 2f þ 1
lnðV f þ1Þ
ð8Þ
The global resistance format introduced to consider the nonlinearity of the limit state function which cannot be considered in the partial safety format [4]. The calculation of design resistance based on the mean value takes the following form:
Rd ¼
lR c~R cRd
ð11Þ
cRd is the model uncertainty factor and takes into account the quality of the resistance model. CEB-FIB code [4] recommends cRd = 1.0 for no uncertainties, cRd = 1.06 for low uncertainties, and cRd = 1.1 for high uncertainties. c~R global resistance factor related to the mean value and can be estimated from full probabilistic analysis. However, for special cases when the distribution of the resistance is known, the global resistance safety factor can be calculated by the method of estimating the coefficient of variation ECOV. In ECOV, the coefficient of variation can be estimated from two non-linear analysis at the mean and the characteristic values of the material strength. If the resistance is normally distributed, the coefficient of variation VR can be calculated as following: VR ¼
1 Rk 1 1:65 lR
ð12Þ
The global central resistance factor can be calculated as follows
c~R ¼
1 1 b aR V R
ð13Þ
If the resistance is log-normally distributed, the coefficient of variation VR can be calculated as following:
Fig. 1. Determination of the characteristic value of the material parameter in case of normal distribution.
T. Bakeer / Structural Safety 62 (2016) 47–56
VR ¼
1 l ln R 1:65 Rk
49
ð14Þ
The global central resistance factor can be calculated as follows
c~R ¼ ebaR V R
ð15Þ
2.3. Full probabilistic approach The design value of resistance is estimated from the probabilistic distribution of the resistance as following:
PðR Rd 6 0Þ ¼ F n ðaR bÞ
ð16Þ
If CDF is the cumulative distribution function of the resistance R, then the design value of resistance is given by:
Rd ¼ CDF 1 fF n ðaR bÞg
ð17Þ
The probabilistic approach can be applied by running a virtual experiment with a large number of random sample data. This procedure is known as Monte Carlo simulation MCS and requires to generate random plausible sampling data of the variables influencing the resistance e.g. material properties. An array of resistance values is generated based on the sampling data, which represent the distribution of the global resistance. The mean, the standard deviation, and the coefficient of variation can be calculated. This fully describes the random properties of resistance. The design value of resistance can be estimated using the reliability index. 3. Nonlinear limit state function
Fig. 2. Cross-section of vertically loaded masonry wall, showing the stress and deformation state.
Since masonry is a material with low tensile strength, the wall may crack under certain conditions leading to further complications due to the reduction in the effective cross-section. Masonry members under compression may fail either because of material over-stressing for squat members or because of stability failure for slender members. For squat masonry members, the failure takes place if the compressive strain at any cross-section reached the ultimate compressive strain of the material. Nevertheless, for slender masonry elements the failure occurs before reaching the ultimate compressive strain of the material at any cross-section. The former mode of failure called material failure and the later called stability failure. Both failure modes are going to play crucial roles in the determination of the capacity reduction factor. For masonry wall with a height h and a thickness t under vertical eccentric loading (Fig. 2), the load bearing capacity is mainly influenced by the eccentricity of the load eo, and the slenderness of the wall k. The slenderness ratio k is defined as following:
k¼
h t
rffiffiffi f E
ð18Þ
where f is the compressive strength of masonry, and E is the initial elastic modulus. The resistance of the wall is defined by the following expression:
NR ¼ f U t
ð19Þ
where NR the resistance load bearing capacity [force/length]. This form has been adopted in many standards to calculate the load bearing capacity. However, more convenient form for the resistance can be defined as following:
nR ¼ f U
ð20Þ
where nR is the strength capacity of the wall due to slenderness and eccentricity. The above equation is useful to demonstrate the influence of the second order effect and eccentricity on the load bearing
capacity by the means of the capacity reduction factor U. A detailed description on determination of the factor U can be found in [7–9]. For short walls, the maximum possible value of nR can be obtained when U = 1. The compressive strength in Eq. (20) can be written with respect to the elastic modulus as following:
f ¼E
k2 r 2h
ð21Þ
where rh = h/t. By substituting in Eq. (20), another demonstration of the stress capacity in term of elastic modulus can be obtained:
nR ¼ E
k2 U r 2h
ð22Þ
For slender walls the maximum possible value of nR can be obtained when:
k2 U ¼
p2 12
ð23Þ
The above value corresponds to the Euler’s buckling load:
nR ¼
E p2 r 2h 12
ð24Þ
The capacity reduction factor has been calculated numerically using the transfer-matrix method described in [10,8]. A graphical representation of the relation between k2 U and k2 at different eccentricities are plotted in Fig. 3. This shows that k2 U reaches a constant value at stability failure. The basic variables of the limit state function for nR are the compressive strength and the elastic modulus:
nR ¼ nR ðf ; EÞ
ð25Þ
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T. Bakeer / Structural Safety 62 (2016) 47–56
Fig. 3. Graphical representation of the relationship between k2 and U k2 showing the areas for material failure and stability failure.
If there is no eccentricity, the limit state function takes the following stepwise form:
nR ðf ; EÞ ¼
8 <
k < 2pp ffiffi3
f;
: rE2 p12 ; 2
h
k P 2pp ffiffi3
ð26Þ
The graphical representation of the limit state function is presented in Figs. 4 and 5 for an example with h/t = 20 and e0/ t = 0.2. It can be noted, from Fig. 5, the stability failure is dominant for bigger values of compressive strength, while the material failure is dominant for bigger values of the elastic modulus. The contours of equal resistances get parallel to f axis at the domain of stability failure. This indicates the independency of the resistance from the compressive strength of masonry at this domain. However, in the domain of material failure, the contours of equal resistances get parallel to E axis only for large values of E. The resistance in this domain is dependent from both E and f. 4. Resistance modelling In many design codes of masonry, the resistance has been modelled considering the uncertainty in the compressive strength of the material. In Eurocode EN 1996-1-1 [12], the resistance of unreinforced masonry subjected to mainly compression action is given by:
NRd ¼ U t f d
ð27Þ
fd is the design compressive strength. By substituting the value of fd from Eq. (2) in Eq. (27), leaves:
NRd ¼ U t g
fk
cm
¼
1
cm
Nðf k Þ
ð28Þ
where N(fk) is a function of the characteristic compressive strength of masonry fk. This shows that the partial safety factor of the material cm has been introduced as a material factor but becomes equal to the global resistance factor as the randomness in U is neglected. Nevertheless, the modelling of the resistance cannot be realistic without considering the uncertainty in the elastic modulus as well. For walls with small slenderness ratio the uncertainty in compressive strength is important but the uncertainty in elastic modulus is not important, on another hand for walls with big slenderness ratio the uncertainty in elastic modulus becomes important for the design while the uncertainty in compressive strength can be neglected. In the following, an example is given considering the uncertainty in the compressive strength but not in elastic modulus. The global safety factor of resistance cR has been calculated using Monte Carlo simulation with one variable for different ratios of h/t. The distribution histograms of the resistance have been demonstrated in Fig. 6. The results show that the partial safety factor cR becomes less for higher slenderness because the influence of the elastic modulus on the resistance becomes less. This shows the importance of using two basic variables (i.e. the compressive strength and elastic modulus) to model the uncertainty in resistance. In the Eurocode 2 (EN 1992-1-1 [13], Section 5.8.6), a partial safety factor has been introduced for elastic modulus to be considered in the design of slender concrete elements.
T. Bakeer / Structural Safety 62 (2016) 47–56
51
Fig. 4. 3D representation of the stress capacity surface in terms of compressive strength and elastic modulus for wall with h/t = 20 and e0/t = 0.2.
Fig. 5. The limit state function of the stress capacity in terms of compressive strength and elastic modulus for wall with h/t = 20 and e0/t = 0.2. The red area represents the practical domain at which the material failure occurs while the blue area represents the practical domain at which the stability failure occurs. The practical domain is considered for compressive strengths f = 2–16 MPa and E/f = 150–2000. In Germany, the practical range of E/f is between 500 for masonry with autoclaved aerated concrete bricks and 2700 for masonry with concrete bricks, (see Table 12 of DIN EN 1996-1-1 NA [11]). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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T. Bakeer / Structural Safety 62 (2016) 47–56
Fig. 6. Histograms for resistance distribution at e0/t = 0.15 generated for a mean compressive strength lf = 5 MPa and considering no uncertainty in the elastic modulus E = 3500 MPa. Two distributions of the compressive strength are considered, (a) normal distribution, and (b) lognormal distribution. The global resistance safety factor is determined by Monte Carlo simulation.
In the following sections, the resistance of masonry has been modelled using two basic variables and the safety factors have been determined for the global safety format and for the partial safety format. 4.1. Global safety format In the global safety format of resistance, the design resistance is given in the following format:
NRd ¼
1
cR
NR ðf k ; Ek Þ
ð29Þ
The joint distribution of the compressive strength f and the elastic modulus E is considered as bivariate normal distribution with no correlation between f and E. The design value of the resistance can be determined by Eq. (16). Considering the nonlinear limit state function of resistance defined in Section 3, the design value of resistance can be estimated by Monte Carlo simulation. The procedure to estimate the global safety factor of resistance can be summarized in the following steps: Step 1: Generation of sampling data for f and E is performed using Latin hypercube sampling method LHS, also known as
T. Bakeer / Structural Safety 62 (2016) 47–56
stratified sampling technique. It represents multivariate sampling method that guarantee non-overlapping designs (Choi et al. [14] and Stein [15]). Step 2: Using the generated sampling data, the resistance can be calculated at each pair of f, E based on the defined limit state function. This gives the probability distribution of resistance. Step 3: Calculation of the design resistance at probability of failure equal to F n ðaR bÞ:
nRd ¼ CDF 1 fF n ðaR bÞg
ð30Þ
Step 4: The global safety factor of resistance can be determined by:
cR ¼
nR ðf k ; Ek Þ nRd
ð31Þ
53
The function nR is demonstrated for an example with e0/t = 0.2 and h/t = 20 together with sampling points generated by LHS method in Fig. 7. The following data were used to generate the sampling points: mean values lf = 5 MPa and lE = 3.5 103 MPa with coefficient of variations Vf = VE = 0.125. The design value of resistance nRd = 1.2 MPa is determined at probability of failure
equal to F n ðaR bÞ ¼ 8:5 105 . The calculated resistance at the characteristic values is nR(fk, Ek) = 1.61 MPa. Consequently, the global safety factor of resistance for this example is cR = 1.35. The same procedure to calculate the global safety factor of resistance is repeated for different values of slenderness ratio and eccentricity. The results are plotted in Fig. 8. It can be noted from Fig. 8 that cR takes the value of 1.5 at extreme values of slenderness i.e. for big and small slenderness ratios. However, cR can reach values less than 1.3 for slenderness k ¼ 0:9 and relative eccentricity e0/t = 0.1. The results were calcu-
Fig. 7. An example showing the determination of the design resistance using sampling data generated by LHS method. The area at which the probability of failure Pf is less than F n ðaR bÞ ¼ 8:5 105 is highlighted in blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. The global safety factor of resistance cR based on the slenderness k and relative eccentricity e0/t estimated by Monte Carlo simulation. The compressive strength and elastic modulus assumed to have equal coefficients of variation Vf = VE = 0.125.
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T. Bakeer / Structural Safety 62 (2016) 47–56
lated assuming equal coefficient of variations for compressive strength and elastic modulus, but if the coefficient of variation for elastic modulus is more than 0.125 it is expected to get values of cR more than 1.5 for big slenderness ratios. 4.2. Partial safety format In the partial safety format, the design resistance is given in the following format:
NRd ¼ NR ðf d ; Ed Þ
ð32Þ
where the design values of the compressive strength and elastic modulus are given as following:
fd ¼
fk
cf
¼
lf Ek l and Ed ¼ ¼ E c~f cE c~E
ð33Þ
cf and cE are the partial safety factors of the compressive strength and elastic modulus respectively with respect to the char~E are the partial safety factors of the com~f and c acteristic values. c pressive strength and elastic modulus, respectively, with respect to the mean values. The partial safety factors can be determined approximately according to the first-order second-moment theory (Melchers [16]) using Hasofer–Lind transformation [17]. The procedure can be described in the following steps: Step 1: Assume initial values of af and aE e.g. af = 1 and aE = 0; Step 2: Calculate the design point on the basis of the assumed a values:
f d ¼ lf ð1 af V f aR bÞ
ð34Þ
Ed ¼ lE ð1 aE V E aR bÞ
ð35Þ
Step 3: Calculation of the partial derivatives of the limit state function at the assumed design point. This can be performed numerically by assuming small values for each of Df and DE in the following equations:
@nR nR ðf d þ Df ; Ed Þ nR ðf d ; Ed Þ ðf ; Ed Þ ¼ @f d Df
ð36Þ
@nR nR ðf d ; Ed þ DEÞ nR ðf d ; Ed Þ ðf ; Ed Þ ¼ @E d DE
ð37Þ
Step 4: Calculation of the sensitivity factors af and aE:
@nR rf @f af ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 @nR R rf þ @n rE @f @E
aE ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a2f
ð38Þ
ð39Þ
Step 5: Checking the newly calculated values of af and aE with the old ones. If the difference is big, the new calculated values should be used again in Step 2; Step 6: Calculation of the partial safety factors with respect to the characteristic values for the compressive strength and elastic modulus:
cf ¼
1 kn V f 1 af V f aR b
ð40Þ
cE ¼
1 kn V E 1 aE V E aR b
ð41Þ
Using the above procedure the partial safety factors has been calculated for an example with h/t = 20 e0/t = 0.3 considering different slenderness ratios. The results are demonstrated in Fig. 9. The results can be explained based on the type of failure. In the area of compression failure, the partial safety factor of compressive strength cf = 1.5 and for elastic modulus cE = 0.8. In the area of stability failure, the values turn to cE = 1.5 and cf = 0.8. For the illustration of these results, three points for the mean values M1, M2, and M3 are considered in Hasofer–Lind transformation space (Fig. 10). The first point M1 chosen in space to get no influence of the elastic modulus on the resistance, while the third point M3 chosen to get no influence of the compressive strength. The second point M2 represent the general case where both elastic modulus and compressive strength contribute to the resistance. At point M1 the design value of the elastic modulus is equal to the mean value Ed1 = lE1. This gives af = 1 and aE = 0. Considering equal coefficient of variation for both compressive strength and elastic modulus Vf = VE = 0.125, the partial safety factors can be obtained directly by applying Eqs. (40) and (41):
Fig. 9. Partial safety factors for the compressive strength and elastic modulus calculated for an example with h/t = 20 and e0/t = 0.3.
T. Bakeer / Structural Safety 62 (2016) 47–56
55
Fig. 10. Geometric interpretation of the possible design points in Hasofer–Lind space.
cf ¼
1 1:65 V f ¼ 1:5 1 0:125 0:8 4:7
cE ¼ 1 1:65 0:125 ¼ 0:8
ð42Þ ð43Þ
At point M3 the design value of the compressive strength is equal to the mean value fd3 = lf3. This gives af = 0 and aE = 1. By applying Eqs. (40) and (41) gives cf = 0.8 and cE = 1.5. Point M2 represents the range of k in Fig. 9 from 0.3 to 0.79. The partial safety factors can be calculated by iteration. 5. Recommendations for non-linear analysis Eurocode EN 1990 gives no regulations on how to deal with safety factors in case of non-linear analysis. Nevertheless, based on the investigation carried out in Section 4.1 on the global resistance safety factor, if the same safety factors of linear limit state functions applied for the non-linear one, it will result in conservative design. Therefore, it is recommended according to Section 4.2 to apply partial safety factors for the compressive strength and elastic modulus, so that the design resistance can be determined as following:
NRd ¼ min½NR ðf d ; lE Þ; NR ðlf ; Ed Þ
ð44Þ
Since the failure state of the structure element is unknown before analysis, it is required to run two non-linear analysis one using the design value of the compressive strength together with the mean value of the elastic modulus, and the other one using the mean value of the compressive strength together with the design value of the elastic modulus. The analysis which gives the minimum capacity is decisive. 6. Concluding remarks In the code of practise of masonry EN 1996-1-1, the resistance of vertically loaded masonry wall has been modelled considering
the uncertainty in the compressive strength of the material. However, for slender masonry walls the uncertainty in the elastic modulus becomes more important than the uncertainty in the compressive strength. The present study suggests modelling the resistance considering the uncertainty in both material parameters, the compressive strength, and the elastic modulus. This is why in EN 1996-1-1, the capacity reduction factor for eccentricity and slenderness U cannot be considered as a constant parameter in the resistance model, but it must be considered as a function of two basic parameters. For a wall with small slenderness ratio, the uncertainty in compressive strength is more important than the uncertainty in elastic modulus. On another hand, for a wall with big slenderness ratio the uncertainty in elastic modulus becomes more important for the design. The partial safety factors are variable and depend mainly on the slenderness ratio of the wall. However, for practical use of partial safety factors, it is suggested to run two non-linear analysis one using the design value of the compressive strength together with the mean value of the elastic modulus, and the other one using the mean value of the compressive strength together with the design value of the elastic modulus. The analysis which gives the minimum capacity is the crucial one.
References [1] Schlune H, Plos M, Gylltoft K. Safety formats for nonlinear analysis tested on concrete beams subjected to shear forces and bending moments. Eng Struct 2011;33(8):2350–6. [2] Schlune H, Plos M, Gylltoft K. Safety formats for non-linear analysis of concrete structures. Mag Concr Res 2012;64(7):563–74. [3] Cervenka V. Global safety format for nonlinear calculation of reinforced concrete. Beton- und Stahlbetonbau 2008;103(S1):37–42. [4] CEB-FIP: Model Code. First complete draft, 1. Lausanne, Switzerland: Internationnal Federation for Structural Concrete; 2010. [5] Allaix DL, Carbone VI, Mancini G. Global safety format for non-linear analysis of reinforced concrete structures. Struct Concr 2013;14(1):29–42.
56
T. Bakeer / Structural Safety 62 (2016) 47–56
[6] EN 1990:2005-12+A1. Eurocode-basis of structural design. European Committee for Standardization; 2005. [7] Bakeer T. Empirical estimation of the load bearing capacity of masonry walls under buckling – critical remarks and a new proposal for the Eurocode 6. Constr Build Mater 2016;113:376–94. [8] Bakeer T. Stability of masonry walls Habilitation thesis. Technische Universität Dresden: Dresden; 2016. [9] Bakeer T, Jäger W, Pflücke T, Christiansen PD. Buckling of masonry with low modulus of elasticity. In: Lourenço Paulo B, Haseltine Barry, Vasconcelos Graça, editors. Proc. of 9th International Masonry Conference. Guimarães, Portugal: Department of Civil Engineering, University of Minho; 2014. paper 1235. [10] Bakeer T. Assessment the stability of masonry walls by the transfer-matrix method. Eng Struct 2016;110:1–20. [11] DIN EN 1996-1-1/NA: 2012–05: Nationaler Anhang – National festgelegte Parameter – Eurocode 6: Bemessung und Konstruktion von
[12]
[13]
[14] [15] [16] [17]
Mauerwerksbauten– Teil 1–1: Allgemeine Regeln für bewehrtes und unbewehrtes Mauerwerk. Deutsches Institut für Normung: Berlin Mai, 2012. EN 1996-1-1:2005-11. Eurocode 6: design of masonry structures – part 1-1: general rules for reinforced and unreinforced masonry structures. European Committee for Standardization; 2005. EN 1992-1-1:2004-12. Eurocode 2: design of concrete structures – part 1-1: general rules and rules for buildings. European Committee for Standardization; 2004. Choi S-K, Grandhi RV, Canfield RA. Reliability-based structural design. London: Springer-Verlag; 2010. Stein M. Large sample properties of simulations using latin hypercube sampling. Technometrics 1987;29(2):143–51. Melchers RE. Structural reliability analysis and prediction. 2nd ed. Chichester, England: John Wiley & Sons; 1999. Hasofer AM, Lind NC. Exact and invariant second-moment code format. J Eng Mech Div 1974;100(1):111–21.