Figure 6.39. Frequencies of relative punching shear carrying capacity
Assuming a normal distribution, a mean value of 0.191 is obtained with a standard deviation of 5 = 0.0247, and a coefficient of variation v = 0.13. Strictly speaking this would mean a characteristic lower bound value of 0.191 - 0.164 ■ 0.0247 = 0.15. assuming a safety factor of 1.5 this would result in a coefficient of 0.10 in stead of 0.12 in the equation of the design punching shear stress (Eq. 6.32b). However, although the variation of the concrete strength (for which a material safety factor 1.5 applies) is the dominating factor with regard to the scatter of results in punching shear tests, the punching shear capacity is not linearly proportional to the concrete compressive strength (the fck value has an exponent 1/3 in Eq. 6.32b ), so that simply applying a material safety factor 1.5 as well for the derivation of the punching shear capacity would be inappropriate. Therefore a more sophisticated approach was necessary in order to unambiguously derive a design equation with the required level of reliability. For such a case the classical "level 2 method", as described in EC-l Basis of Design is suitable. The way how to deal with this method has been described and illustrated by Taerwe . The same method was applied by Konig and Fischer for investigating the reliability of existing formulations for the shear capacity of members without shear reinforcement .
According to the level 2 method, a reliable design equation can be derived from test results with the general formulation
Mbr mean value of test results br sensitivity factor for Br, normally taken as 0.8 in the case of one dominating parameter p target safety index, taken 3.8 5br coefficient of variation with mbr = 0.191, qbr = 0.8 and 5br = 0.130 a value for the design coefficient in Eq. 6.32b of 0.116
was obtained. In this derivation, however, the mean concrete cylinder compressive strength has been used, whereas in the code expression the 5%-lower value fck is used. In the Model Code the relation fck = fcm - 8 (MPa)
is given. This means a coefficient of variation for a concrete C25 of v = 0.15 and for a concrete C90 of v = 0.05. In combination with Br = 0.13 for the punching tests this would mean an increase of the coefficient 0.116 of about 6.8 % for C25 and 3.6 % for C90 (see [14, p. 91). This would then result in a coefficient 0.124 for a concrete class C25 to 0.120 for a concrete class C90. It can therefore be concluded that equation 6.32a,b is correct.
A disadvantage of Eq. 6.32.a,b is that they go to 0 if p goes to 0. This could give unrealistically low punching shear capacities for low reinforcement ratio's, that may for instance occur in prestressed slabs. Furthermore designers like to have a simple lower bound formulation for a fist check.
Therefore a lower bound was added, solely depending on the concrete tensile strength, according to the relation vu > C- fctd, where fctd = fctk /yc. Evaluating the same results as shown in Fig. 6.38 with the equation vu = C- fctk, it was found that for normal strength concrete (<C50/60) a 5% lower value C =0,57 applied and for high strength concrete (>C50/60) a value equal to C= 0,42. It should however be noted that the collection of tests does not contain slabs with cross-sectional depths larger than 275 mm. In order to cope with larger slab depths used in practice the coefficient C 5% should therefore be further reduced. The value 0,35, used as a lower limit for shear as well, seems to be quite reasonable. Taking into account this lower li the design equation for non-prestressed slabs should therefore be:
184.108.40.206 Punching shear resistance of prestressed slabs without shear reinforcement.
For shear loaded members the influence of a normal compression force is taken into account a separate contribution of 0,15acp to the ultimate shear stress VRd.c (prENV 1992-1-1:2001 6.2a), see also the report for shear. It is logic that also with regard to punching the effect of prestressing will be positive. It is however not expected that the same term as for shear can be used, because the contribution of prestressing to the punching resistance depends also decisively on the definition of the control perimeter.
In order to find the contribution prestressing a selection of test results has been made: Andersson , Gerber & Bums , Stahlton , Pralong, Brandli, Thurlimann  and Kordina, Nolting . A comparison of the test results with the design equation:
VRd,c = (0,18/yc) k (100pi fck)1/3 - 0,08ocp > 0,35 fctk /yc - 0,08oc
The mean value of vexp/ vRd,c is 1,58 and the standard variation is s = 0,20. According to Eq. 6.33, with a = 0,8 and ( = 3,8 the design value should be (1,58 - 0,8 . 3,8 . 0,20) vRd,c = 0,972 vRd,c. Actually this means that Eq. 6.36, giving values which are only slightly (2,8%) too high, is acceptable as a design equation.
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