## MviAxi max xi

is used in the dimensioning of the shear reinforcement of DCH beams as a measure of the reversal of the shear force at end i (similarly at end;').

The design shear force in primary seismic columns and beams of buildings of DCM or DCH is always computed through equations (D5.12) and (D5.13), without exemptions. In beams and columns with short clear length ld, these expressions give a large value of the design shear force. Short columns are very vulnerable to the high shear force resulting from equation (D5.12), and special precautions should be taken at the conceptual design stage to avoid them. In short beams the last term in equations (D5.13) is small, and equation (D5.14) gives a value of (, close to -1. Although not so problematic as short columns, short beams are difficult to dimension for the high shear force from equation (D5.13a) and for a value of Q close to -1. (cf. Section 5.7.6, p. 122). So they should also be avoided, through proper spacing of the columns. It is noted at this point that although coupling beams of shear walls may be short, they are subject to special dimensioning and detailing rules to ensure ductile behaviour under their high and fully reversing shear forces.

Clauses

The values of theMRd b in equations (D5.12) and (D5.13) can be computed from equations (D5.1) and (D5.2), and'those of MRd c from equations (D5.3) and (D5.4). MRi c should be computed for the value of the axial load that is most unfavourable for the verification in shear. For the columns, as

• the shear resistance increases with the value of the axial load (both the shear resistance controlled by transverse reinforcement, KRd s, and that controlled by diagonal compression in the web of the member, KRd max, cf. Section 5.7.6) and

• the shear force demand from equation (D5.12) increases with the moment resistance of the column, MRd c, and in turn MRd c increases when the axial load increases up to the balance load (i.e. the load at which crushing at the extreme compression fibres takes place is exactly when the tensile reinforcement reaches its yield stress)

the most unfavourable of the following two cases should be considered,

(1) the minimum value of column axial forces in the seismic design situation from the analysis

(2) the value of the axial load, within its range of variation in the seismic design situation, for which MRd c becomes a maximum. This is the value of MRd c computed for the minimum of the following two values: the maximum value of the column normalized axial load in the seismic design situation, vmiX, and the balance load, vh,

The value of MRd c for the balance load vb, can be computed from equation (D5.2) with £ taken as

The variables in equations (D5.15) and (D5.16) are as defined for equations (D5.3) and (D5.4), and are computed using the design values/yd and/cd as fy and fc, respectively; the conventional values, ec2 = 0.002, ecu2 = 0.0035, are used as ec and ecu, respectively, in equations (D5.15) and (D5.16).

The axial load in beams is normally zero, so the values of MRd c in equations (D5.13) should be the maximum ones determined according to point 2 above.

When the value of the design shear force from equations (D5.12) and (D5.13) is so high that it exceeds the shear resistance, as this is controlled by diagonal compression (web crushing), then it will normally be more effective for the eventual fulfilment of the verification of the beam or column in shear to reduce its cross-sectional dimensions, than to increase them. The member flexural capacity, MRd, that determines to a large extent the magnitude of the design shear force from equations (D5.12) and (D5.13) is more sensitive to the cross-sectional dimensions of the member than its shear resistance, as this is controlled by diagonal compression, VRd max. This is more so when the member longitudinal reinforcement is controlled by minimum requirements, or if the change in cross-sectional dimensions has a more-than-proportional effect on the moments (from the analysis) for which the longitudinal reinforcement is proportioned (this is normally the case in columns exempt from the satisfaction of equation (D4.23) and in beams with reinforcement at the supports controlled by the seismic design situation and not by vertical loads).

5.6.4.3. Capacity design shear force in ductile walls

Ductile walls are designed to develop a plastic hinge only at the base section and to remain elastic throughout the rest of their height. The value of the flexural capacity at the base section of the wall, MRd0, and equilibrium alone are not sufficient for the determination of the maximum seismic shears that can develop at various levels of the wall, because, unlike in the beam of Fig. 5.5, the horizontal forces and the moments applied to the wall at floor levels are not constant but change during the seismic response. In the face of this difficulty a first assumption is that if MRdo exceeds the bending moment at the base as obtained from the elastic analysis for the design seismic action, MEdo, seismic shears at any level of the wall will exceed those from the same elastic analysis in proportion to MRdo/MBdo. So, the shear force from the elastic analysis for the design seismic action, VEd, is multiplied by a capacity design magnification factor £ which takes up the following values:

In buildings of DCH:

• for 'squat' walls (those with a ratio of height to horizontal dimension, hjlw, < 2):

• for 'slender' walls (those with a ratio of height to horizontal dimension, hjl^, > 2):

Edo J

sjjJ

In buildings of DCM: • for simplicity:

The value of e from equations (D5.17) and (D5.18) should not be taken greater than the value of the q factor, so that the final design shear, VEd, does not exceed the value qVEd corresponding to fully elastic response. Moreover, it should not be taken as less than the constant value of 1.5 provided for DCM.

As described in Section 5.8.3, £ values higher than those of equations (D5.17) and (D5.18) are specified for large lightly reinforced walls, which are always designed for DCM, and are often squat.

The factor 1.2 in equations (D5.17) and (D5.18) attempts to capture the overstrength at the base over the design value of the flexural capacity there, MRdo, e.g. owing to strain hardening of vertical steel. In the second term under the square root sign of equation (D5.18), SJT^ is the value of the elastic spectral acceleration at the period of the fundamental mode in the horizontal direction (closest to that) of the wall shear force which is multiplied by e, and SC(TC) is the spectral acceleration at the corner period, Tc, of the elastic spectrum. This latter term aims at capturing the increase of shear force over the elastic overstrength value represented by the first term, due to higher-mode effects in the elastic and the inelastic regime of the response, after a proposal by Eibl and Keintzel.63 In modes higher than the first one, the ratio of the shear force to the bending moment at the base exceeds the corresponding value at the fundamental mode considered to be primarily (if not exclusively) reflected by the results of the elastic analysis. The longer the period of the fundamental mode, the lower the value of Sc(Tt) and the higher that of e, reflecting the more significant effect of higher modes on the shears. It should be pointed out, though, that equation (D5.18) has been proposed as a correction factor primarily on the results of the 'lateral force' (equivalent static) procedure of analysis for the design seismic action. If the elastic analysis is indeed dynamic ('modal response spectrum' analysis), then its results reflect the effects of higher modes on - at least the elastic - seismic shears.

Higher-mode effects on inelastic shears are larger in the upper storeys of the wall, and indeed more so in dual structural systems. The frames of such systems restrain the walls in the upper storeys, and the shear forces at the top storey of the walls from the 'lateral force procedure' of elastic analysis are opposite to the total applied seismic shear, becoming zero one or two storeys below. Multiplication of these very low storey shears by the factor e of equations (D5.16)-(5.18) will not bring their magnitude anywhere close to the relatively high

storey shears that may develop there due to higher modes (cf. dotted curves representing the shear forces from the analysis and their magnification by e in Fig. 5.6). In the face of the unrealistically low magnified shear forces in the upper storeys, Section 5 requires that the minimum design shear of ductile walls in dual systems is at the top at least equal to half of the magnified shear at the base, increasing linearly towards the magnified value of the shear, sVEd, at one third of the wall height from the base (Fig. 5.6).

If the axial force in the wall from the analysis for the design seismic action is large (e.g. in slender walls near the corner of high-rise buildings, or in coupled walls), there will be a large difference between the absolutely maximum and minimum axial force in the individual walls in the seismic design situation (including the axial force due to gravity loads). As the vertical reinforcement at the base of the wall is controlled by the case in which the bending moment from the analysis, MEdo, is combined with the minimum axial compression, the flexural capacity when the maximum axial compression is considered at the base, MRdo, is much larger than MEd0. Then, the value of s from equation (D5.17)maybeso high that the verification of the individual walls in shear (especially against failure by diagonal compression) may be unfeasible.

5.6.4.4. Capacity design shear in beam-column joints Clause 5.5.2.3 Unlike gravity loading, which normally induces bending moments in beams which are of the same sign at opposite sides of a joint, seismic loading induces very high shear forces in beam-column joints. The magnitude of the shear in a joint can be appreciated if that joint is considered as part of the beam and it is noticed that the beam bending moment changes from a (high) negative value to a positive one across the joint, producing a vertical shear force, V-, equal to the average of the product of the seismic shear force in the beams, Vb, and their clear span, Lbn, divided by the column depth, hc. Similarly, if the joint is considered as part of the column, the change in the column bending moment from a high value at the face of the joint above to an equally high value of opposite sign at the face below produces a horizontal shear force, Vjh, equal to the average of the product of the seismic shear force in the columns above and below the joint, Vc, and their clear storey height, hstn, divided by the beam depth, hb. These shear forces correspond to a nominal shear stress in the concrete of the joint equal to the ratio of Y,MC = ^Mb to the volume of the joint, taken equal to hjibbwhere bj is the effective width of the joint, taken according to Section 5 as if bc > bw then b-s = min{bc; (bw + 0.5hc)}; otherwise fy = min{bc; (bv + 0.5hc)} (D5.20)

Shear stresses are introduced into a joint mainly through bond stresses along the beam and column bars framing the core of the joint. Because the nominal shear stress in the concrete of the joint is the same, regardless of whether it is computed from the horizontal or the vertical shear force, Vjh or Vjv respectively, from the capacity design point of view it is more convenient to compute it from Vjh based on the forces transferred via bond stresses along the top bars of the beam, as beams - even those not fulfilling equation (D4.23) -normally yield before the columns (this is on the safe side for the joint, even if beams do not yield). If bond failure along the top bars of the beam does not occur, the maximum possible value of Vjh can be computed as the sum of the maximum possible tensile force in the top bars y4sb/y on one side of the joint plus the maximum possible compressive force in the top flange on the opposite side, minus the shear force Vc in the column above the joint. Irrespective of how it is shared by the concrete and the top reinforcement, the maximum possible compression force in the top flange will be controlled by the bottom reinforcement, and will be equal to its maximum possible tensile force, Asb2fy. Therefore, the design value of the horizontal shear force in the joint is

where the beam reinforcement is taken at its overstrength, 7Rd/yd, and the shear force Vc in the column above may be taken equal to the value from the analysis for the seismic design situation. It is obvious from the derivation of equation (D5.21) that in the sum (Aibl +/lsb2) the top beam reinforcement area,^4sbl, refers to one vertical face of the joint and the bottom one, Asb2, to the opposite face, so that the larger of the two sums should be considered. Normally, though, no such distinction needs to be made, especially as in interior joints the same bar area is provided at either side of the joint. At exterior joints only one term in the sum (Asbl +Asb2) should be considered.

Equation (D5.21) is applied with an overstrength factor of 7Rd= 1.2 for beam-column joints of DCH buildings. For simplicity, in DCM buildings the beam-column joints are not dimensioned in shear on the basis of the shear force computed from equation (D5.21) but are treated through prescriptive detailing rules that have proved fairly effective in protecting joints in past earthquakes.

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