## Dimensioning of shear reinforcement in critical regions of beams and columns

Clauses The design value of the shear resistance of beams or columns is computed according to the

5.4.3.1.1(1), rules of Eurocode 2 for monotonie loading, both when it is controlled by the transverse 5.4.3.2.1(1), reinforcement, VRd s, and when it is controlled by diagonal compression in the web of the 5.5.3.2.1(1) member, FRd max. There is one exception to this: the value of FRd s in the critical regions of beams of DCH. The special rules for FRd s in this particular case are described below. Clause In the critical regions of beams of DCH the strut inclination, 9, is taken equal to 45°

5.5.3.1.2(2) (cot 9 = 1). This is equivalent to a classical Môrsch-Ritter 45° truss with no concrete contribution term (Vcd = 0). The underlying reason is the experimentally observed reduction of VRd s in plastic hinges (i.e. after flexural yielding) with the magnitude of inelastic cyclic deformations. In members that have initially yielded in bending, this reduction manifests itself by a rapid increase of shear deformations with load cycling, leading ultimately to shear failure. This phenomenon is described conveniently and fairly accurately on the basis of a classical Môrsch-Ritter 45° truss model for shear resistance under cyclic loading with non-zero concrete contribution term, Vc, considering that either the Vc or the sum of Vc and the contribution of transverse reinforcement, Vw, decrease with the plastic part of the imposed displacement ductility factor, pf=p6-l. The models developed for concrete beams, columns (rectangular or circular) and walls in Biskinis et al.69 and adopted in Annex A of EN 1998-352 are of this type (in units of meganewtons and metres):

FR,S =^min (N; 0.554/c) + (l-0.05min(5; pf))[Vw +VC] (D5.36a)

VKs = ~^-min(N; 0.55AJC) + VW +(1-0.095min(4.5; pf))Vc (D5.36b)

where:

• x is the compression zone depth

• N is the compressive axial force in the seismic design situation (positive, zero for tension)

0 Ls, equal to M/V, is the shear span at the end of the member end

• Ac is the cross-section area, equal to b„d for a cross-section with a rectangular web of thickness bw and structural depth d, or to ttDc2/4 (where Dc is the diameter of the concrete core to the inside of the hoops) for circular sections

• the concrete contribution term is equal to

Fc = 0.16max(0.5; 100ptot) l-0.16minj^5; ^jjTcA (D5.37)

where ptol is the total longitudinal reinforcement ratio 0 the contribution of transverse reinforcement to shear resistance is equal to:

(a) for cross-sections with rectangular web of width (thickness) b%y:

where:

pw is the transverse reinforcement ratio

- z is the length of the internal lever arm (z = d-d' in beams, columns, or walls with a barbelled or T section, z = 0.8/w in rectangular walls)

- / is the yield stress of transverse reinforcement

(b) for circular cross-sections

where:

- D is the diameter of the section

- Asw is the cross-sectional area of a circular stirrup

- s is the centreline spacing of stirrups

- c is the concrete cover.

In buildings designed for plastic hinging in the beams, the value of /./,„ in these beams is normally equal to the global displacement ductility factor, p,s, that corresponds to the value of q used in the design via equations (D2.1) and (D2.2). Therefore, depending on the value of ajax and the regularity classification of the building, the value of pf ranges from 1.5 to 3.5 in DCM beams and from 2.5 to 5.5 in DCH ones. According to equation (D5.38a), the ensuing reduction of VR s is small for DCM beams, but may be significant in DCH ones. For simplicity, in the case of DCM beams the reduction is neglected, and the normal expression for VRds from Eurocode 2 is applied (that expression employs only the Vv term from equations (D5.38), multiplied by cot 9, with cot 0 between 1 and 2.5). For DCH beams, where the reduction of VR s with ¡if in plastic hinges cannot be neglected, and as in the context of Eurocode 2 no Vc term is used in the expression for Vm s, cot 0 = 1 is taken (cf. equations (D5.38)), which is equivalent to a reduction of Vc to zero, instead of the reduction by about half suggested by equation (D5.36b). Figure 5.8, in which the test data used to fit equations (D5.36) and (D5.37) have been cast in the format of a model with variable strut inclination, 9, shows by how much this approximation is conservative.

Plastic hinging is not expected in the columns of dissipative buildings designed to Eurocode Clause

8. If it does take place, it will normally lead to lower chord rotation ductility demands and less 5.5.3.2.1(1)

ensuing reduction of the value of VR s than in beams. It is expected that, if such a reduction occurs, its effects will be offset by the 7Rd factor of 1.1 for DCM and of 1.3 for DCH employed in the capacity design calculation of shear force demands (cf. equation (D5.12)). So, for columns the reduction of shear resistance in plastic hinges is neglected, and the normal expression for KRd s from Eurocode 2 is applied. That expression employs the Vv, term from equations (D5.38), multiplied by cot 0, with cot 9 between 1 and 2.5, as well as the contribution of the inclined compression chord given by the first term in equations (D5.36) without the 0.55AJC upper limit. For simplicity, that term may be taken as equal to (d - djllcl.

Clause The second point where the shear verification of plastic hinges in DCH beams deviates

5.5.3.1.2(2) from the Eurocode 2 rules refers to the use of inclined bars at an angle ±a to the beam axis against sliding in shear at the end section of the beam. Such sliding may occur in an instance when the crack is open throughout the depth of the end section and the shear force is relatively high. For this to happen, a significant reversal of the shear force is necessary, as well as a high value of the peak shear force. A value of ( from equation (D5.14), which is algebraically less than -0.5, is the criterion adopted in Section 5 for a significant reversal of the shear, and a value of the maximum shear from equation (D5.13a) greater than (2 + QfaJ),xd is the limit for a peak shear capable of causing sliding for ( <-0.5. This limit shear is between one-third to one-half of the value of VRdimax for cot 9 = 1. As the surface susceptible to shear sliding is not crossed by stirrups, if these limits are exceeded, inclined bars crossing this surface should be dimensioned to resist through the vertical components Asfyd sin a of their yield force - in tension and compression - at least 50% of the peak shear from equation (D5.13a). The 50% value corresponds to the limit value (= -0.5, and respects the recommendation of clause 9.2.2(3) in Eurocode 2 to resist at least 50% of the design shear through links. If the beam is short, the inclined bars are most conveniently placed along its two diagonals, as in coupling beams; then, tan a~(d- d')/lcl. If the beam is not short, then the angle a of the diagonals to the beam axis is small, and the effectiveness of inclined bars placed along them is also low; two series of shear links, one at an angle a = 45° to the beam axis and the other at a = -45°, would be effective then. The construction difficulties and reinforcement congestion associated with such a choice are obvious, though. Normally there is neither risk from sliding shear nor a need for inclined reinforcement, if the configuration of the framing is selected to avoid beams that are relatively short and are not loaded with significant gravity loads in the seismic design situation (i.e. having a high value for the first term and a low value for the second term on the right-hand side of equations (D5.13)).

Clause Plastic hinges in columns are subjected to an almost full reversal of shear (( ~ -1), and the

5.5.3.2.1(1) peak value of the shear force from equation (D5.12) is normally high. However, no inclined

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Fig. 5.8. Experimental data on the dependence of the strut inclination 9 on the imposed chord rotation ductility ratio, for cyclic loading after flexural yielding69

bars are required for them, as, due to the axial force and the small magnitude of plastic strains in vertical bars, the crack is expected to always be closed over part of the depth of the end section. Moreover, sliding is resisted through clamping and dowel action in the large-diameter vertical bars, which are normally available between the extreme column reinforcement in the section and remain elastic at the instance of peak positive or negative response of the column. Verification against sliding shear and placement of inclined bars to resist it is required, though, in ductile walls of DCH (see Section 5.7.9), as in walls the axial load level is lower and the web bars are of smaller diameter and more sparse than in columns. An important practical difference between columns and walls in this respect is that, due to the size, density of transverse and longitudinal reinforcement and one-directional nature of the cross-sectional shape and function of the walls, inclined bars can be easily placed and are quite effective in shear; this is not the case in columns, on exactly the same grounds.

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