Verification of beamcolumn joints in shear
Clauses Assuming that bond strength along the beam and column bars framing the joint core is
5.5.3.3(I), sufficient to transfer into the joint the full shear force demand, given by equation (D5.20) in
5.5.3.3(2), terms of the horizontal shear force, Vjhd, the body of the joint then resists that shear. This
5.5.3.3(3) shear force is translated into a shear stress, considered uniform within the joint volume, defined by the horizontal distance between the extreme layers of column reinforcement, h, the net depth of the beam between its top and bottom reinforcement, hyA„ and the (horizontal) width, b, of the joint given by equation (D5.20):
There is no universally accepted rational model for the mechanism through which the joint resists cyclic shear and ultimately fails. Experimental results on interior joints collected and compiled by Kitayama et al.66 suggest that the joint shear resistance, expressed in terms of the shear stress, v, of equation (D5.29), increases about linearly with the ratio of horizontal reinforcement within the joint, pih, from v ~ 0.15/c for pjh = 0 (unreinforced joint) to a limit value between v] ~ 0.24fc and Vj = 0.4/c (mean value: Vj = 0.32/c) at pjh = 0.4%. Above that value of the steel ratio and up to pjh = 2.4%, ultimate strength seems to always be attained by diagonal compression in the concrete and to be practically independent of the value of pjh and of the axial load ratio in the column, v = N/fcAc.
Guided by the test results mentioned above and in view of the lack of consensus on models, Section 5 has adopted a very simple plane stress model for the verification of the shear strength of beamcolumn joints in DCH buildings. The model assumes homogeneous stresses in the body of the joint, consisting of:
(1) the shear stress, vj; from equation (D5.29)
(2) the vertical normal stress, N/Ac = z/c = vdfci (compression), from the column
(3) a horizontal normal stress, p^f^ (compression), as a reaction to the tensile force that develops in the horizontal reinforcement when the latter is driven to yielding by the dilatancy of the joint at imminent failure.
Joint strength criteria are based on the principal stresses, in tension, cr„ and compression, £7n, under the system of stresses 13 above. The required ratio of horizontal reinforcement, pjh, is obtained from the condition that a, does not exceed the concrete tensile strength,/ct, vi2
/ct+f/c or, using the design values of the strengths, including/ctd =/ctk 0 0S/7C = 0.7fctJ%,
bih)v, /ctd+^d/cd where Ash denotes the total area of the horizontal legs of hoops within the joint, between the top and bottom reinforcement of the beam. For a safesided (conservative) estimate of/lsh, vd in equation (D5.30b) is computed from the minimum value of the axial force of the column above the joint in the seismic design situation. It is noteworthy that for pjh  0 equation (D5.30a) gives values of Vj ranging from 0.1/c to 0.2fc for values of u between 0 and 0.3, in good agreement with the average value of v ~ 0.15/c suggested for p]U = 0 by the compilation of test results by Kitayama et al.66
The other verification condition is that au does not exceed the concrete compressive strength, as this is reduced due to the presence of tensile stresses and/or strains in the transverse direction (i.e. that of u,). The reduced compressive strength is taken to be equal to r]fcd = 0.6(1 ~/ck(MPa)/250)/cd (the reduction factor 77 is the same as factor v applied on/cd in clause 6.2.3 of Eurocode 2 for the calculation of the shear resistance of concrete members, as this is controlled by diagonal compression in the concrete; the symbol 77 was used in Eurocode 8, to avoid confusion with the frequently used normalized axial load v). The adverse  effect of the horizontal normal stress, p^f^, on the magnitude of <rn, as well as its (more important) favourable effect on the compressive strength in the diagonal direction through confinement, are both neglected. So the condition rjfcd < au gives
Equation (D5.31) is the verification criterion of interior beamcolumn joints against diagonal compression failure. At exterior joints we rely on 80% of the value in equation (D5.31):
Unlike equations (D5.30), where for the verification to be safesided (conservative) the minimum value of the column axial force in the seismic design situation should be used, equations (D5.31) and (D5.32) should employ the maximum value of the column axial force in the seismic design situation (including the effect of the overturning moment in exterior joints). For common values of vd (—0.25), equation (D5.31) gives values of the shear stress, vj; close to 0.4/cd, which is at the upper limit of the strength values compiled by Kitayama et al.66 for interior joints. Experimental results suggest that an ultimate value of the shear stress, Vj, close to 0.4/cd can be attained in columns with a slab at the level of the top of the beam and a transverse beam on both sides of the joint. For exterior joints, which are normally checked with a higher value of vd due to the effect of the overturning moment on column axial force, equation (D5.31) gives results close to the mean experimental value of 0.32fcd observed in interior joints without transverse beams and a top slab. The conclusion is that, unless the value of /cd =/ck/7c uses a partial factor for concrete, (significantly) higher than 1.0, equations (D5.31) and (D5.32) do not provide a safety margin against failure of the joint by diagonal compression.
As an alternative to equations (D5.30), Section 5 derives the joint horizontal reinforcement Clause 5.5.3.3(4) from a physical model proposed by Park and Paulay.67 According to that model, a joint resists shear via a combination of two mechanisms:
(1) a diagonal concrete strut between the compressive zones of the beams and columns at opposite corners of the joint
(2) a truss extending over the entire core of the joint, consisting of
(any) horizontal hoops in the joint
(any) vertical bars between the corner bars of the column (including column longitudinal bars contributing to the flexural capacity of the end sections of the column above and below the joint) a diagonal compression field in the concrete.
The force in the strut is assumed to develop from:
• the concrete forces in the beam and column compression zones at the two ends of the strut
• the bond stresses transferred to the joint core within the width of the strut itself.
The truss resists the rest of the joint shear force. Then, for the dimensioning of the horizontal joint reinforcement to be safesided (conservative), the horizontal component of the strut force should not be overestimated. With this in mind, the assumption in Paulay and Priestley68 is adopted, namely that at the face of the joint where the beam is in positive bending (tension at the bottom) the crack cannot close at the top flange, due to accumulation of plastic strains in the top reinforcement. This is very conservative for the truss and its horizontal joint reinforcement, because the compression zone of the beam does not deliver a horizontal force to the concrete strut, but only a compressive force to the beam top reinforcement to be transferred (together with the tension force at the opposite face of the joint) to the truss and the strut, in proportion to their share in the joint width at the level of the top reinforcement. As the horizontal width of the strut at that level is equal to the depth of the compression zone of the column above the joint, xc, and assuming  for simplicity that the transfer of the total force (Asbl +Asb2)fy by bond takes place uniformly along the total length, hc, of the top bars within the joint, a fraction of this force equal to xjh,. goes to the horizontal force of the strut, and the rest, (1 xjhc), to the truss. It is both realistic and safesided for the truss horizontal reinforcement to consider that the column shear force, Vc, appearing as the last term in equation (D5.21) for VjM, is applied directly to the strut through the compression zone of the column above and affects only its horizontal shear force, not that of the truss. So, as the whole depth of the vertical faces of the joint are taken up by the truss, the total area, Ash, of the horizontal legs of hoops within the joint should be dimensioned for the force (1 xc/hc)(Asbl +Asb2)fy. The value of xjhc may be computed from equation (D5.3), using lo1 = to2, = 0 (for convenience), eco = 0.002 and ecu = 0.0035 (for spalling of the extreme concrete fibres at the end section of the column). Then, 4 = z/,/0.809 = vj{l.5 x 0.809) = 0.8z/d, with both u(S and £c normalized to hc. So, the following total area of horizontal hoops should be provided:
8 At interior joints,
where 7Rd is taken equal to 1.2 (as in equation (D5.21) for DCH) and the normalized axial force ud is the minimum value in the column above the joint in the seismic design situation.
Reinforcement requirements at exterior joints cannot be obtained by setting Asb2 = 0 in equation (D5.33). The underlying reason is that the beam top reinforcement is bent down at the far face of the joint, and when it is in tension it delivers at the bend to the diagonal strut, starting there the full diagonal compression force of the strut. The horizontal component of that force is close to fyAsbl  Vc, and so very little force is transferred by bond along the part of the top bars outside the strut, to be resisted as horizontal shear by the truss between the strut and the face of the joint towards the beam. What governs the horizontal shear force of the truss is the force transferred by bond along the part of the bottom bars outside the strut (the upward bend of bottom bars at the far face of the joint does not deliver forces to the joint core when these bars are in compression). The compression zone at the bottom of the beam delivers to the bottom end of the strut a horizontal force equal to the compression force in the concrete, i.e. equal to the difference between the tension in the top reinforcement, Asblfy, and the force in the bottom reinforcement, which yields in compression, Asb2fy. The difference between the horizontal component of the strut force at its top end,^4sbl/y  Vc, and the horizontal forces delivered at its bottom end from the beam and the column below, (Asbl Asb2)fy  Vc, and by bond within the strut width at the level of the bottom reinforcement, (1  xJhc)Asb2fy, is the force transferred by bond along the part of the bottom bars outside the strut width and to be to be resisted by the truss as horizontal shear between the strut and the external face of the joint. This gives:
8 At exterior joints, where again 7Rd = 1.2, butvA is the minimum value of the normalized axial force in the column below the joint in the seismic design situation.
The two alternative models, equations (D5.21) and (D5.30) and equations (D5.33) and (D5.34), give quite dissimilar results. The amount of reinforcement required according to equations (D5.21) and (D5.30) is very sensitive to the values of ;/d and v (suggesting that according to this model the shear resisted by means of the diagonal tension mechanism is insensitive to the amount of horizontal reinforcement), whereas the joint reinforcement required according to equations (D5.33) and (D5.34) is rather insensitive to the value of vA and proportional to v. For mediumhigh values of vA (around 0.3) equations (D5.21) and (D5.30) require much less joint reinforcement than equations (D5.33) and (D5.34), whilst for low values of vd (around 0.15) equations (D5.21) and (D5.30) require less joint reinforcement than equations (D5.33) and (D5.34) for v < 0.3/cd, and the opposite if v > 0.3/cd. For nearzero values of vA, equations (D5.21) and (D5.30) require much more joint reinforcement than equations (D5.33) and (D5.34), especially for high values of vr If this discrepancy is disturbing, even less reassuring is the difference between the predictions of either model and the experimental strength values compiled by Kitayama et al.66 for interior joints: for a given shear stress demand, the experimental evidence is that much less joint reinforcement is needed than given by either of the two models. The only case of acceptable agreement with the test results is that of equations (D5.21) and (D5.30) for mediumhigh values of vA (around 0.3). The conclusion of these comparisons is that the designer may use with confidence the minimum of equations (D5.21) and (D5.30) and equations (D5.33) and (D5.34) for the steel requirements.
The truss mechanism underlying equations (D5.33) and (D5.34) includes as one of its Clauses components vertical reinforcement that provides the vertical tensile field equilibrating the 5A.3.2.2(2), vertical component of the diagonal compression field in the concrete. Intermediate bars 5A.3.2.2(I l)(b), between the corner ones, arranged along the sides of the column with depth hc, can play that 5A.3.3(3), role, along with contributing to the flexural capacity of the end sections of the column above 5.5.3.2.2(2), and below the joint. Such bars are provided along the perimeter at a spacing of not more 5.5.3.2.2(I2)(c), than 150 mm for DCH or 200 mm for DCM, to improve the effectiveness of concrete 5.5.3.3(9), confinement. For the present purposes, Section 5 requires at least one intermediate vertical 5.5.3.3(5)
bar between the corner ones, even on short column sides (less than 250 mm for DCH or 300 mm for DCM).
For the joints of DCH buildings, where the horizontal joint reinforcement area,/4sh, needs to be calculated through equations (D5.21) and (D5.30) or equations (D5.33) and (D5.34), the total area of column intermediate bars between the corner ones, Asv; ¡, should be
determined from^4sh as follows:
The coefficient j accounts for the normally smaller inclination of the strut and the truss compression field to the vertical, compared with the diagonal of the joint core. It also limits the effect of the overestimation of Ash by equations (D5.21) and (D5.30) or equations (D5.33) and (D5.34) in affecting the vertical reinforcement as well. Clauses The computational verification of beamcolumn joints according to equations (D5.30)
5.4.3.3(I), (D5.35) is required only in DCH buildings. For DCM, the detailing measures prescribed by
5.4.3.3(2), Section 5 for both DCH and DCM joints without any calculation suffice. According to these
5.5.3.3(7), measures, the transverse reinforcement placed in the critical regions of the column above or
5.5.3.3(8) below (whichever is the greatest) should also be placed within the joint, except if beams frame into all four sides of the joint and their width is at least 75% of the parallel crosssectional dimension of the column. In that case the horizontal reinforcement in the joint is placed at a spacing which may be double of that at the columns above and below, but not more than 150 mm.
To see what the prescriptive rules above imply for the minimum horizontal reinforcement in the joint, it is recalled that for DCH the critical regions of columns above the base of the building should be provided with a minimum design value of 0.08 for the mechanical volumetric ratio of transverse reinforcement, wwd. For S500 steel and concrete grade C30/37, this value corresponds to pjh  0.185% per horizontal direction if the partial factors for steel and concrete are equal to their recommended values for the persistent and transient design situations, or to pjh = 0.24% if they are set equal to the recommended value of 1.0 for the accidental design situation (for other concrete grades the minimum value of ph is proportional to /c). Although other constraints on the column transverse reinforcement in critical regions (e.g. that on the diameter and spacing of transverse reinforcement: dbh > max(6 mm; 0.4<ihI ), sw < min(6dbL; bJ3; 125 mm), or on the minimum value of pglt ensures) may govern, it is indicative that the values quoted above for pjh are well below the value of 0.4% that marks the limit of the contribution of horizontal reinforcement to the shear resistance of the joint according to Kitayama et al.(* For DCM, Section 5 has no lower limit on iowd in the critical regions of columns, only a limiting hoop diameter (dbb > max(6 mm; dbJ4)) and spacing (yw < min(8dbL; bJ2; 175 mm)). These limit values give a low horizontal reinforcement ratio in the joint. Considering that the practical minimum for DCM is 8 mm hoops, with a horizontal spacing for the legs of 200 mm, at a hoop spacing of 125 mm, the resulting steel ratio in the joint is p]b = 0.2% per horizontal direction.
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