Figure 11.3. Verification of Eg. 11.8 for the shear capacity of members without shear reinforcement with test results

According to the level 2 method, a reliable design equation can be derived from test results with the general formulation


BRd design value Pbr mean value of tests a,BR sensitivity factor for Br normally taken 0.8 in the case of one dominating parameter p target safety index, taken 3.8 5br coefficient of variation with pbr _ 0.162, aBR _ 0.8 and 5br _ 0.145 a value for the design coefficient in Eq. 11.8 of 0.091 is 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 new version of EC-2, according to the Model Code, the relation fck _ fcm - 8 (Mpa) (11.10)

is used. This means coefficients of variation 5f _ 0.15 for a concrete LC 25/30 and 5f _ 0.055 for a concrete LC 80/95. This would mean an increase of the coefficient 0.091 with 9% for C25 and 3% for LC 80/95.This would then result in a coefficient 0.100 for a concrete class LC25/30 and 0.095 for a concrete class of LC80/95. The conclusion is that Eq. 11.8 should be modified to

V1Rdc _ [0.10 m k (100p fck)1/3 - 0.15ocd] bwd (11.11)

This agrees with the proposal given in [2].

Shear capacity of members with shear reinforcement

In the new version for EC-2, contrary to ENV 1992-1-1, only one method for the design of members with shear reinforcement is given. This method is based on the variable angle truss model. For members of normal density concrete not subjected to axial forces, with vertical shear reinforcement, the shear capacity is the smaller value of

where Asw _ cross-sectional area of one stirrup, s _ stirrup distance, z _ inner lever arm of the cross-section, fywd _ design yield stress of shear reinforcement, 9 _ inclination of compression strut, and

the additional condition a f

The first eq. 11.12a represents yielding of the shear reinforcement and the second equation 11.12b crushing of the inclined concrete struts. The inclination of the concrete struts can freely be chosen between 21.80 (cot 0 = 2.5) and 450 (cot 0 = 1).

v is an efficiency factor for the concrete crushing strength depending on the concrete strength according to:

An important question with regard to the applicability of those formulations for lightweight aggregate concrete is if the concrete struts in the web have a sufficient capacity to rotate. In normal density concrete during crack formation the strong aggregate particles do not fracture and the crack propagates around them: therefore the crack surface is very rough so that large frictional forces can be transmitted. This is a very important condition to allow a rotation of the inclined struts from 450 down to an angle of 21.8° as a minimum. In lightweight aggregate concrete the aggregate particles are intersected, so that a less rough crack surface is obtained. It is therefore questionable whether the rotation capacity of the web is sufficient to allow as well a lowest strut inclination of 21.8°, or if a higher lower limit should be defined. In order to answer this question tests have been carried out on I-shaped beams with varying shear reinforcement, Fig. 11.4, Walraven [21]. Any series consisted of three beams, which contained whether low, medium or high ratio's of shear reinforcement, whereas furthermore the beams were exactly similar. Three types of lightweight aggregates were used in the various concrete mixes: Lytag, Liapor and Aardelite. In those concretes only the coarse aggregate particles were of the lightweight type: the mixtures contained natural sand. The concrete volume weights were 2050 kg/m3 (Aardelite), 1975 kg/m3 (Lytag) and 1780 kg/m3 (Liapor). Those series were compared with a reference series with beams made of normal density concrete. On the web the state of deformation was continuously measured, so that the inclination of the principal compression strain could be monitored. Fig. 11.4 shows two diagrams, in which the inclination of the principal strain is represented. The left diagram shows the results for the gravel concrete members, with low (GD30L), medium (GD30M) and high (GD30H) shear reinforcement ratio's, the right diagram shows the corresponding curves for lightweight concrete. The tests show that the rotational behaviour of the inclined struts is similar for LWAC and NDC. In both cases the beams with the lowest shear reinforcement ratio showed the highest strut rotation capacity. Obviously the other two shear reinforcement ratio's were both too high to reach yielding of the steel, so that the final rotation remained relatively small. The unexpected result that NDC- and LWAC-beams behave similarly can be explained by the overall shape of the cracks. On a meso-level the roughness of the cracks in lightweight concrete is indeed smaller, but this was compensated by the roughness on the macro-level, caused by the overall crack undulation. In this way also in the interface contact areas occurred, with sufficient capacity to develop the necessary transmission of forces across the inclined cracks. Also Thorenfeld [22] reported a substantial decrease of the strut inclination with increasing load. For a shear reinforcement ratio of 0.5% he found a lowest strut inclination at failure of 250. His tests were carried out on lightweight concrete with Leca aggregates, both for the course and the sand fractions. The volume weight of this concrete was 1500 kg/m3.

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