One variable action

Results of the reliability analyses are presented in graphical form that indicates variation of the reliability index failure probability Pf, and sensitivity factors aR, aE, aG, aQ and aW with the load ratio x In particular Figure 3 shows results of a simple case of one variable action only (the main variable action Q); Figure 3 indicates the variation of

- the reliability index

- failure probability Pf, and

- for expression 6.10 of EN 1990 sensitivity factors aR, aE, and partial sensitivity factors aG, aQ and aW

with the load ratio x

For the analysis it has been assumed that a single variable action, the imposed load Q having the characteristic given in Table 3 is acting on the generic element only (i.e. k = 0.0). A middle value for the global safety factor yR = 1,15 and for coefficient of variation VR = 0,15 have been considered.

Element Sensitivity Factor
Figure 3. Variation of the reliability index P, the failure probability Pf and the sensitivity factors aR, aE, a, ocq and aW with the load ration x for k = 0, for a generic cross-section assuming yR = 1,15 and the coefficient of variation VR = 0,15.

It follows from Figure 3 that for the assumed higher coefficient of variation VR = 0,15 only the combination rule A (i.e. expression (6.10) of EN 1990) [1] seems to be fully acceptable (P > 3.8 and Pf.< 7,23x 10-5) in the interval 0 < x < 0.8, however the reliability level considerably varies with x In some cases the alternative A might lead to an uneconomic design.

The alternative B (i.e. expression (6.10a) and (6.10b) of EN 1990) is acceptable in a slightly shorter range of x, 0 < x < 0 7 than the variant A but provides obviously much more uniform distribution of reliability level with x. Obviously it would lead to a more economic design than the alternative A. Alternative C (i.e. modified expression (6.10a) and (6.10b) of EN 1990 [1]) is providing rather low reliability level particularly for the interval 0 < x < 0.3 and should not be used unless partial factors yare changed.

Similar results were obtained in previous studies [11,12,13,14] of structural elements made of different materials (concrete and steel elements). These studies differ from the presented results primarily by the value of the partial factor yR and the coefficient of variation VR (and also by the asymmetry of the distribution of R). Just the conclusions formulated above seem to be supported by a number of different material oriented examples.

Note that the sensitivity factor aR increases to about aR ~ 0,9 while the factor aE decreases, aE > - 0,5, indicating that the resistance gives a greater contribution to safety than intended by EN 1990. However this conclusion is strongly dependent on assumed coefficient of variation VR. With increasing VR the sensitivity factor aR increases. It is interesting to note that than the sensitivity factors are very close to the values recommended in EN 1990 [1], i.e. aR ~ 0,9 aE ~ -0,7.

5.2 Two variable actions

A more general case when two variable actions (a leading imposed load Q, together with an accompanying action W) are acting is shown in Figure 4, which (similarly as Figure 3) shows the variation of the

• reliability index ft,

• failure probability Pf, and

• for expression 6.10 of EN 1990 sensitivity factors aR, aE, aG, ocq and aW with the load ratio x for k = 0,75 and the coefficient of variation VR = 0,15.

The case considered for Figures 3 (i.e. k = 0, with a single imposed load Q acting) is extended so that a more detailed insight of the effect for the reliability parameters considered can be obtained. However Previous investigations [11,12] clearly show that reliability in case of two variable actions is considerably greater than reliability in case of one variable action.

I

1 1 1 1

; aw aQ '■

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i

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Load ratio CHI=(Qk+Wk)/(Gk+Qk+Wk)

Figure 4. Variation of the reliability index ft, the failure probability Pf and the sensitivity factors aR, aE, a, ag and aW with the load ratio x for k = 0.75, for a generic cross section assuming yR = 1,15 and the coefficient of variation VR = 0,15.

It follows from Figure 4 that for the assumed coefficient of variation VR = 0,15 and the consideration of two variable actions the reliability of the generic cross-section exposed to two variable actions is considerably greater than the reliability of the same cross-section exposed to one variable action only. This finding also indicates that the factor y/W may be rather high. Note that the sensitivity factors aR seems to be slightly greater than the values aR = 0,8 considered in EN 1990 [1] and aE in absolute value is less than aE = - 0,7 recommended in [1]. This finding depends on assumed variability of basic variables.

Figure 5. Variation of the reliability index P with the load ratio x and partial factor for resistance yr and k = 0 (i.e. imposed load Q is the only variable action), for the generic-cross section assuming partial safety factors yG = 1,35 and y/Q = 1,5, and the coefficient of variation

Figure 5. Variation of the reliability index P with the load ratio x and partial factor for resistance yr and k = 0 (i.e. imposed load Q is the only variable action), for the generic-cross section assuming partial safety factors yG = 1,35 and y/Q = 1,5, and the coefficient of variation

It follows from Figure 5 that for the assumed variables the acceptable domain of the load ratio x and the coefficient of variation VR is limited by the contour line determined as an intersection of the P surface and the plain P= 3,8 in Figure 5. Obviously with increasing yr reliability index P increases, yR = 115 would be satisfactory for most of the practical range of the load ratio x (for the load ratio x< 0,8).

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