O beams without stirrups • beams with stirrups Aiwfuw / bs = 0.58 MPa g|-1-1-1-1-

Figure 6.16. Results of tests on beams subjected to axial tension, bending and shear, and failing in shear [Regan, 1999]

For similar arguments, reference is made to [Bhide and Collins, 1989] Shear tension capacity

In special cases, like for instance when pretensioned strands are used in members with reduced web widths, such as in prestressed hollow core slabs, shear tension failures can occur, Fig. 6.17.

Figure 6.17. Shear tension failure

In this case failure occurs due to the fact that the principal tensile stress in the web reaches the tensile strength of the concrete in the region uncracked in flexure. The principal tensile strength in the web calculated using Mohr's circle, Fig. 6.18, is equal to i . rrrr

substituting % = VRd,ct S/bwI and on = ai Ocp the code's expression EC-2, Eq. 6.3 V =ilbs

Rd,ct s is obtained. Loads near to supports

In 6.2.2 (5) the equation (5) or in prENV 1992-1-1:2001 the Equation 6.2.a, is extended with a factor (2d/x) in order to cope with the increased shear capacity in the case of loads applied near to supports. According to this formulation, at a distance 0.5d < x < 2d the shear capacity may be increased to

VRd,ct = 0,12 k (100 Pl fck)1/3 (2d/x) bw d. (6.18)

This may need some explanation, since it might be argued that loads near to supports may be treated with the rules given in EC-2, 2001 version, chapter 6.5 "Design of discontinuity regions with strut and tie models".

However, there are many arguments in favour of the formulation according to Eq. 6.18: - According to the formulations for the strut and tie model the capacity of the concrete struts only depends on the strength of the concrete, see e.g. fig. 6.19.

Consequently, the maximum capacity is a function of the concrete strength and the width of the support area.

Figure 6.19. Bearing capacity of short member according to strut and tie model with defined maximum concrete stress in the struts

It can easily be seen that this is a very simplified representation of reality, since the capacity of such a member results to be independent of the slenderness ratio a/d, which is known to have a strong influence. Furthermore short members are prone to significant size effects. It was shown [Lehwalter and Walraven, 1994), that the size effect in short members is the same for short and slender members, so that here also the factor k = 1 + V(200/d) applies.

Lehwalter carried out tests on short members with various sizes, a/d ratio's and support widths, and compared the equivalent maximum stress in the concrete struts, Fig. 6.20. The dotted plane is valid for a maximum stress 0,6 fc. It is seen that for lower a/d ratio's the capacity is considerably higher than the one obtained with the strut and tie model. It is seen furthermore that the limit 0.55 fc as defined in 6.4.5 5(P) for struts with transverse tension is appropriate for a/d < 2.0, members with depths until 1 m and a support width up to about 0.25d.

For a number of practical members, like in the case of corbels and pile caps, it is important to reduce the size as much as possible. A more accurate formulation than the strut and tie model is therefore useful in those cases.

- Another case is shown in Fig. 6.21. It is a part of a foundation caisson in the Storebaelt bridge, with a slab of about 1 meter and wall distances of about 5 m. A substantial part of the counterpressure of the soil is transmitted directly to the walls, so that the governing shear load is small. Without a provision like the one given in Eq. 10, unnecessary shear reinforcement would be required.


Figure 6.20. Maximum stress in concrete struts as calculated on the basis of test results (Walraven, Lehwalter, 1989)
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