## Maximum diameter of longitudinal beam bars crossing beamcolumn joints

Shear forces are introduced to beam-column joints primarily through bond stresses along Clause 5.6.2.2(2)

the beam and column longitudinal bars framing the joint core. Equation (D5.21) above giving the design shear force in the joint presumes that bond strength along the beam top bars is sufficient for the transfer of this shear force. Although loss of bond along these bars will not have dramatic global consequences, it would be better avoided through verification of bond along the bars of the beam. This verification has the form of an upper limit of the diameter of the longitudinal bars of the beam, dbL, that pass through interior beam-column joints or are anchored at exterior ones. This upper limit is derived as follows.

If 1 and r (denoting 'left' and 'right') index the two vertical faces of the joint, <rs is the stress in the beam bars, and if hco is the width of the confined core of the joint parallel to the depth hc of the column, then the average bond stress along these beam bars is

.._ ra^bL 1 asi - gs2 1 = dbb I Osi - gS2 1 (D5 25)

b 4 irdbLhco 4 hco with bond stresses along the length of the bars outside the confined core considered negligible. Plastic hinges are assumed to develop in the beam at both the left and right faces of the joint. As the top flange is normally much stronger than the bottom flange both in tension and in compression, its force cannot be balanced unless the bottom bars yield. So, in the bottom bars we have crs, = -/ and <rs r = /, and rb is equal to dbLfyl2hco. Regarding the top bars, it is assumed that at beam plastic hinging they yield at the face at which they are in tension: as ,=/ . At the right face of the joint their compressive stress, as r, is such that, together with the force of the concrete in the top flange, Fc r (negative, as compressive), it balances the tension force in the bottom bars. These latter bars have a cross-sectional area As r2, and at plastic hinging they are forced by the stronger top flange to yield, so that cr.

where />, and p2 are the ratios of top and bottom reinforcement at the right face normalized to the product bd of the beam, u is defined as lo = p jy/fc and £rff is the depth of a fictitious compression zone, normalized to d, such that Fc r = -bd£eiifc. Therefore, at the top bars Tb is equal to and its value is lower than along the bottom bars for the same value of dhL. However, the bond problem seems to be more acute along the top bars, because bond stresses are not uniformly distributed around the perimeter of the bar but are concentrated more on the side facing the joint core. At the top bars this is the underside of the bar, where bond conditions are considered 'poor' due to the effects of laitance and consolidation of concrete during compaction. At the bottom bars bond conditions are considered 'good'.

According to Eurocode 2 the design value of the ultimate bond stress is 2.25/ctd for 'good' bond conditions, and 70% of that value for 'poor' conditions. The design value of the concrete tensile strength is /ctd =/ctk 0.o5/7c= 0.7/ctm/7c. As the consequences of bar pull out from the joint core will not be catastrophic (it will increase the apparent flexibility of the frame and the interstorey drifts and it may prevent the beam from reaching its full flexural capacity at the joint face), basing the design bond strength on the 5% fractile of the tensile strength of concrete and - in addition - dividing it by the partial factor for concrete seems unduly conservative. So, this partial factor is not applied here. As bond outside the confined joint core is neglected, the positive effects of confinement by the joint stirrups, the top bars of the transverse beam and the large volume of the surrounding concrete are considered according to the CEB/FIP Model Code 90,64 i.e. by doubling the design value of the ultimate bond stress instead of dividing it by 0.7 according to Eurocode 2. The result for the top bars ('poor' bond conditions), equal to 0.7 x 2.25 x 0.7/ctm x 2 = 2.2/ctm, may be increased by the friction due to the normal stress on the bar-concrete interface, a cos2 tp, produced by the mean vertical compressive stress in the column above the joint, a = NEi/Ac = z//cd. Using the design value /./, = 0.5 specified in Eurocode 2 for the friction coefficient on an interface with the roughness characterizing that between the concrete and the bar and integrating the friction force /j.a cos2 ip around the bar (i.e. between tp = 0 and 180°), friction increases the design value of bond strength to 2.2/ctm + 0.5 x 0.5z/d/cd = 2.2/ctm(l + 0.8z;d). The factor 0.8 in parentheses incorporates a value of 10.5 for the ratio of/ck = 1.5/cd to/ctm (this ratio varies between 9 and 11.8 for C20/25 to C45/55, and the value of 10.5 corresponds to C30/37). Setting rb from equation (D5.27) equal to this design value of bond strength along the top bars, the following condition is derived for the diameter of beam longitudinal bars in beam-column joints, dbL:

8 in interior beam-column joints

8 in beam-column joints which are exterior in the direction of the beam i + o.8i/d) (D5.28b)

h-c 7Rd/yd where the overstrength coefficient for the beam bars, 7Rd, is taken to be equal to 1.0 for DCM

and to 1.2 for DCH. The coefficient k represents (1 - £eti/uj) in equation (D5.27); in equation

(D5.28a) the coefficient is taken equal to k = 0.5 for DCM and to k = 0.75 for DCH. In exterior beam-column joints we have as2 = 0, which is equivalent to k = 0, giving equation

(D5.28b). The value of vd=NEd/fcàAc should be computed from the minimum value of NEd in

the seismic design situation; although no special instructions are given in Eurocode 8 for tensile net axial forces (as may occur in exterior columns of medium- or high-rise buildings), it is clear from the way equations (D5.28) are derived that in that case = 0.

It is most convenient to apply equations (D5.28) at the stage of initial sizing of columns, on the basis of the desired maximum value of beam bar diameter. This can be done on the basis of a rough estimate of the minimum axial load ratio v6 in the seismic design situation (corresponding to only the gravity loads in interior columns and gravity minus axial forces due to the overturning moment in exterior ones). At that stage the final value of the top reinforcement ratio p1 in equation (D5.27) will not be known, so in equation (D5.28a) the value of px in equation (D5.27) was taken equal to the maximum value allowed, px max, from equation (D5.24). At the same stage the bottom steel ratio p2 may be taken to be equal to the minimum value from equation (D5.22), or to 0.5p! max. These convenient choices for p2 and Pi, max are unconservative for db] . This should be viewed, though, bearing in mind that equation (D5.28a) is very demanding for the size of interior columns: a column size hc of over 40dbh is required for DCH, common values of axial load (vA ~ 0.2), steel with nominal yield stress of 500 MPa and relatively low concrete grade (C20/25) - i.e. hc over 0.6 m if dbL =14 mm and over 0.8 m when dbL = 20 mm. The requirement is relaxed to about 30dbh for medium-high axial loads and higher concrete grades. If DCM is chosen, the required column size is reduced by about 25%. Although onerous, such requirements are justified by tests: cyclic tests65 on interior joints show that the cyclic behaviour of beam-column subassemblages with hc = 18.75dbL is governed by bond slip of the beam bars within the joint and is characterized by low-energy dissipation and rapid stiffness degradation; a column size of hc = 31.5dbh was needed for the cyclic behaviour of the subassemblage to be governed by flexure in the beam and to exhibit stable hysteresis loops with high energy dissipation (subassemblages with hc = 28dbL gave intermediate results). According to Kitayama et al.,66 the energy dissipated by subassemblages with hc - 20dbL cycled to a storey drift ratio of 2% corresponds to an effective global damping ratio of only 8%.

Although equation (D5.28a) has been derived for the top bars, according to Eurocode 8 it applies to the bottom bars of the beam as well. For the bottom bars the denominator in the second term of equation (D5.28a) should be replaced by 2, and term 7.5/ctm in the numerator should be divided by 0.7, to account for the 'good' bond conditions. The end result is about the same as that from equation (D5.28a), so for simplicity the same expression is used for bottom bars as well. It should be noted, though, that for the bottom bars of exterior joints, equation (D5.28b) is conservative by a factor of about 0.7 for the required column depth hc due to 'good' bond conditions.

For exterior joints, equation (D5.28b) is conservative for both the top and bottom bars for another reason: although at the exterior face of such joints, top beam bars are normally bent down and bottom bars up, equation (D5.28b) takes into account bond only along the horizontal part of these bars and discounts completely the contribution of the 90° hook or bend. Underpinning this are Table 8.2 and clause 8.4.4 of Eurocode 2, according to which only the straight part of the bar counts toward anchorage in compression. The potential of push-out of 90° hooks or bends, if the straight part of the bar is not sufficient to transfer the full bar yield force to the joint, was also behind the adoption of Eurocode 2 in this respect. However, 90° hooks or bends near the exterior face of such joints are protected from push-out - as well as from opening up and kicking out the concrete cover when in tension -by the dense stirrups placed in the joint between the 90° hook or bend and the external surface. Moreover, top bars are normally protected from yielding in compression by the overstrength of the top flange relative to the tensile capacity of the bottom flange. So, only the bottom bars may yield in compression at an exterior joint; but for them the margin of about 0.7 for hc noted above is available. The same margin of about 0.7 for hc is available according to the Eurocode 2 rules for anchorage in tension of top bars with a 90° standard hook or a bend near the exterior face of the joint. On these grounds, 70% of the value of hc required by equation (D5.28b) may be used at exterior joints, without reducing their safety against bond failure below that provided by equation (D5.28a) for interior ones. Section

Anchor plate oL > 0.6cL

- Hoops around column bars

Fig. 5.7. Detailing arrangements in exterior beam-column joints proposed in Section 5 as an alternative to straight anchorage of beam bars

5 proposes the anchorage arrangements in Fig. 5.7 as an alternative to increasing the column size or reducing the diameter of beam bars in exterior beam-column joints to meet equation (D5.28b).

Equations (D5.28) lead to the use of square columns in two-way frames. Moreover, unless column sizes are large for other design reasons (drift control, strong column-weak beam design to satisfy equation (D4.23), etc.), equations (D5.28) lead also to small diameters of beam bars. To prevent them from buckling, such bars need to be restrained by closely spaced stirrups, especially at the bottom of the beam which lacks the lateral restraint provided at the top by the slab.

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