Coefficient Of Variation On Annual Maximum Snow Uk Met Office

Figure 1. Variation of pn with p1 for n = 5, 25, 50 and 100

Note that, if 1-year period would be used for specification of the target reliability level of a structure, then Figure 1 provides information on the resulting failure probability corresponding to a given working life Tn. For example, if the target reliability level is specified by the reliability index p1 = 4,7 (corresponding to the probability p1 = 1,3 x 10-6), then (as already mentioned) the reliability level of a structure having a working life, Tn = 50 years is characterised by p50 = 3,8. Similarly when a period Tn = 5 years is used, then p5 = 4,3 or when Tn = 100 years, then p100 = 3,6.

So, the reliability level of a structure can be specified using different time periods T, which may not necessarily coincide with the design working life Td. This may be useful when experimental data concerning time variant basic variables are available for a specific reference period T (for example 1 or 5 years) that is different from the design working life Td. In such a case, however, all the basic variables (including those that are time independent) should be considered by appropriate design values related to the same reference period T. The following simple example indicates the effect of using a reference period T different from the design working life considering a resistance variable (strength) having lognormal distribution.

5 PARTIAL FACTOR OF A MATERIAL PROPERTY

Consider a resistance variable R (strength) having lognormal distribution. When an alternative reference period Ta instead of the design working life Td is used in reliability verification of a structure, then the design value of R should be determined for Ta instead of Td. It is assumed that the characteristic value Rk of R is defined as its 5% fractile [5], [6] a [7].

Then of the resistance variable R, the characteristic value Rk and design value Rd are defined as [4], [5]

Taking into account equations (8) and (9) it follows that the partial factor is given as

Yr = Rk / Rd = exp (- 1,645 x Vr) / exp (-or x pa x Vr) (10)

Considering selected values of the coefficient of variation VR, Figure 2 shows the partial factor YR for lognormal distribution of R (equation (10)).

It follows from Figure 2 that when reliability of a structure is verified using a short alternative reference period Ta (for example for example for Ta = 1 year when pa = 4,7), the partial factor yr should generally be greater than in the case when the whole design working life Td (for example for Td = 50 when pd = 3,8) is considered. It may be noted that the partial factor yr of material property R increases with the increasing value of the reliability index pa.

Similar conclusions can be expected for partial factors of other basic variables, in particular for partial factors of permanent actions.

Lognorm Variation
Figure 2. Variation of YR with pa for selected coefficients of variation (R lognormal)

6 PARTIAL FACTORS OF SELF-WEIGHT

Consider a self-weight G having normal distribution. Similarly as in the case of material property, when an alternative reference period Ta instead of the design working life Td is used in reliability verification of a structure, then the design value of G should be determined for Ta instead of Td. The characteristic value Gk of G is defined as the mean mg [5], [6] and [7]:

Gd = Ug - oco x p x gg = m + 0,7x pa x gg = Ug(1 + 0,7x Pa X Vg) (12)

In equation (11) and (12) jg denotes the mean, gg the standard deviation, VG the coefficient of variation and aG = - 0,7 the sensitivity factor of G. The partial factor yG of G is defined as [5], [6] a [7]

Taking into account equations (11) and (12) it follows from (13) that

Figure 3 shows variation of the partial factor Yo with the reliability index pa for selected values of the coefficient of variation VG = 0,05; 0,10; 0,15 and 0,20. Note that YG = 1,35 (recommended in EN 1990 [5]) corresponds approximately to the reliability index pa = 3,8 if the coefficient of variation is about 0,1 (the value in EN 1990 [5] was increased by 5% to take into account model uncertainty).

Figure 3. Variation of yG with pa and coefficient of variation VG (G normal).

Assuming the coefficient of variation 0,1 for both the resistance R and the self weight G Figures 2 and 3 indicate that the partial factor of self-weight yG varies slightly more significantly with pa - values than with the partial factor yR of resistance variable R. This finding is, however, dependent on the distributions assumed for both variables.

7 CLIMATIC ACTIONS AND IMPOSED LOADS

Drafts of European documents for climatic actions due to temperature [8], snow [9] and wind [10] indicate possible reduction of characteristic values Qk for temperature, snow load and wind speed in case of shorter reference (return) period (for example 5 years) than 50 years considered in normal cases. Such a reduction may be applied in transient design situations (for example during execution).

The following relationships for thermal, snow and wind actions, respectively, are recommended in relevant Parts of Eurocode EN 1991:

(a) In accordance with EN 1991-1-5 [8] Thermal actions, the maximum and minimum shade air temperature Tmax,50/Tmin,50 for 50-year return period may be reduced to Tmax,n/ Tm\n„ for n-year return period using the following formulae

Tmin,n = k Tmin,5o, for k = {£3 + £4 ln[-ln(1-1/n)]}

where Tmax,n/Tmin,n is the maximum/minimum, and the coefficients k1 = 0,781, k2 = 0,056, k3 = 0,393, k4 = -0,156 might be used (based on data of UK [11]),

(b) In accordance with EN 1991-1-3 [9] Snow actions the characteristic value of snow action sk,n corresponding to the return period of n years is given using Gumbel distribution as

1 - V — [ln(- ln(1 - p)) + 0,57722] sk,n = k sk,50, where k =-n-

where sk,50 is the characteristic snow load on the ground for 50-year return period and sk,n for «-year return period, p denotes here the probability of sk,n being exceeded corresponding to n years of return period and Vs is the coefficient of variation of annual maximum snow load,

(c) In accordance with EN 1991-1-4 [10] the basic wind speed vb,n having the return period n years may be assessed using semi-empirical expressions vb,n = k vb,50, where k =

where vb,50 is the basic wind velocity for 50-year return period and vb,n for n-year return period and p denotes here the probability of vb,n being exceeded corresponding to n years of return period. The constant K in equation (18) follows from Gumbel distribution as K = Vv-vé/n, where Vv denotes coefficient of variation of annual wind speed. An approximate value K = 0,2 (which corresponds to the coefficient of variation Vv= 0,26) is used in the following comparison of reduction coefficients k for considered climatic actions.

Table 3 shows reduction coefficients k for climatic actions (applied in a general relationship Qk,n = k Qk,50) for selected return periods of n - years.

Table 3. Reduction coefficient k for climatic actions (Qk,n = k Qk,50) for different return periods of n - years.

Return period

P

Reduction coefficient k for

of n-years

1 max,«

T

sn,n

Vb,n

2 years

0,5

0,8

0,45

0,64

0,77

5 years

0,2

0,86

0,63

0,75

0,85

10 years

0,1

0,91

0,74

0,83

0,90

50 years

0,02

1

1

1

1

It follows from Table 3 that the characteristic value of climatic actions may be considerably reduced if shorter reference period is considered in the design. For example for 5-year return period of action due to snow or wind reduces to 75 or 85 % of the characteristic values for 50-year return period, similarly the characteristic value of the maximum shade air temperature to 86%, the minimum shade air temperature even to 63%. Note that in verification of bridge decks during execution phases the characteristic values of uniform temperature components are derived from shade air temperature [8].

It should be noted that no reduction of partial factors for load is indicated in documents [8], [9] and [10]. Thus, the same reliability level as for 50-year design working life described by pd = 7,2 x 10- 5 (fid = 3,8) may be considered also for the reference period T = n years. Certainly, a different reliability level (for example reduced to fid < 3,8) can be chosen taking into account economic and other aspects in accordance to the principles of reliability differentiation discussed above.

Imposed load could be possibly also reduced when short reference time is considered similarly as climatic actions. Some statistical data are available in documents of JCSS [12]. However, a variety of random properties of different types of imposed loads make it very difficult to formulate general rules. Unless convincing data are available the characteristic values specified in current documents may be accepted without any reduction.

8 EXAMPLES

Consider a steel structure having the design working life Td = 50 years, for which the target failure probability is specified aspd = 7,2 x 10- 5 (fid = 3,8). Failure probability p for the alternative reference period Ta = 1 year, which is considered in design due to data concerning actions, will be lower than the target failure probability pd (p < pd and fi> fid); from equation (6):

When the reference period Ta = T1 = 1 year is considered in design verification, then the reliability index fi follows from equation (7) as fix = - 0-1(1,44 x 10- 6) = 4,7

Reliability index fi1 is greater than the target value fid = 3,8 specified for the design working life Td = 50 years.

Using equation (10) the partial safety factor yR for Ta = T1 = 1 year assuming the coefficient of variation VR = 0,08 (corresponding to the common variability of strength of structural steel) the partial safety factor is given as (see also Figure 2)

Yr = exp (- 1,645 x 0,08) / exp (- 0,8 x 4,7 x 0,08) = 1,18

Note that when the design working life Td = 50 is considered in reliability verification then:

Yr = exp (- 1,645 x 0,08) / exp (- 0,8 x 3,8 x 0,08) = 1,12

Obviously, the partial factor yr increases with the decreasing reference period Ta.

The partial factor of self-weight yg is given by equation (14). Assume again, that the specified reliability level for 50-year design working life is given by fid = 3,8. Assuming the coefficient of variation VG = 0,1 and considering the one year time period for reliability verification (fi1 = fia = 4,7), then the partial factor yg that should be used is

If the verification period is equal to the design working life (pd = pa = p50 = 3,8), then

Thus, the variation in yG is less significant than the variation in yR (see also Figure 3).

A different task is reliability verification of an agricultural structure having the design working life Td = 25 years, for which the target reliability index can be decreased to pd = 3,3 (see Table 1). It follows from equation (10) that the partial factor yR for Td = 25 is

Yr = exp (- 1,645 x 0,08) / exp (- 0,8 x 3,3 x 0,08) = 1,08

The partial factor yR may, therefore, be decreased from 1,15 to about 1,1. However it should be emphasized that this reduction of yR is due to a reduced target reliability index pd = 3,3 and not due to a shorter design working life Td = 25 instead of the usual Td = 50 years.

Annex A includes MATHCAD Sheet "GammaRG" that can be used to make numerical calculations.

9 CONCLUDING REMARKS

(1) In present international documents the target values of failure are related to economic aspects of safety measures and consequences of structural failure only vaguely, without any explicit relation to various design working lives Td for different types of structures.

(2) When alternative failure probability pa is derived for a suitable reference period Ta from the target failure probability pd and design working life Td, partial factors and characteristic values of variable actions for pa and Ta should also be specified.

(3) For temporary structures, with a short design working life Td, the target failure probability pd can be specified in accordance with the general principles of reliability differentiation; reliability elements for basic variables should be derived for specified pd and Td.

(4) The partial factors y derived for an alternative reference period Ta different from Td may vary considerably from the values corresponding to the design working life Td depending on Ta and distributions of relevant basic variables.

(5) The partial factor of self-weight yG corresponding to an alternative reference period Ta varies with pa-values less significantly than the partial factor of material property yR.

(6) Partial factors yR derived for an alternative reference period Ta of one-year may be considerably greater than yR specified for the design working life Td.

(7) Following recommendations of Eurocodes, the characteristic value for climatic actions due to snow corresponding to 5-year return (reference) period may be reduced to 75 % of the characteristic values for 50-year return period, similarly the characteristic value of wind speed may be reduced to 85 %, the maximum temperature to 86%, the minimum temperature to 63%.

REFERENCES

[1] Gulvanessian, H. - Calgaro, J.-A. - Holicky, M.: Designer's Guide to EN 1990, Eurocode: Basis of Structural Design; Thomas Telford, London, 2002, ISBN: 07277 3011 8, 192 pp.

[2] CSN 730031 Structural reliability. Basic requirements for design. (Spolehlivost stavebnich konstrukci a zakladovych pud. Zakladni ustanoveni pro vypocet), CSNI 1990.

[3] CSN 731401 Design of steel structures. (Navrhovâni ocelovych konstrukci), CSNI 1998.

[4] ENV 1991-1 Basis of design and actions on structures. Part 1: Basis of design. CEN 1994.

[5] EN 1990 Eurocode - Basis of structural design. CEN 2002.

[6] ISO 2394 General principles on reliability for structures, ISO 1998.

[7] ISO 13822. Basis for design of structures - Assessment of existing structures, ISO 2001.

[8] EN 1991-1-5 Eurocode 1 Actions on structures. Part 1-5: Thermal actions. CEN, 06/2002.

[9] EN 1991-1-3 Eurocode 1 Actions on structures. Part 1-3: Snow actions. European Committee for Standardisation, 06/2002.

[10] EN 1991 Actions on structures. Part 1-4: Wind load. European Committee for Standardisation, 06/2002.

[11] G. König et al: New European Code for Thermal Actions, Background document, Report N.6, University of Pisa, 1999.

[12] JCSS: Probabilistic model code. JCSS working materials, http://www.jcss.ethz.ch/, 2001.

ATTACHMENTS

1. MATHCAD sheet "GammaRG.mcd"

MATHCAD sheet Gamma is intended for determination of the partial factor yR of the resistance R and the partial factor yG of the permanent load G.

2. MATHCAD sheet "PSIO.mcd"

MATHCAD sheet PSI0 is intended for determination of the Combination factor for accompanying action.

3. MATHCAD sheet "PSI12.mcd"

MATHCAD sheet PSI12 is intended for determination of the combination factor y12 for accompanying action.

Attachment 1 - MATHCAD sheet "Gammarg.mcd"

GammaR, gammaG for a theoretical model

MATHCAD sheet for determination of the charactreistic, design values and partial factors y R and y G. Coeficients of fractile estimation given in EN 1990

Sensitivity factors: aR := 0.8 aE :=-0.7 pR(p):=P aR pE(p):=P aE

Characteristic and design values (relative values related to the mean ) xk=^kCT*pX, xd=^ds*^x gjn(V) := (1 - k-V) §dn(p, v) := (1 - pR(p)-v) |^dn(3.8,0.1) = 0.696 |

Normal distribution

Normal distribution Çkln(V) :=

GammaR

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