## Implementation of capacity design of concrete frames against plastic hinging in columns

5.6.2.1. The left-hand side of equation (D4.23)

Clause 5.2.3.3(2) The design value of the flexural capacity of a beam in negative (hogging) bending may be where Asl and As2 (Asl >As2) are the cross-sectional areas of the top and bottom reinforcement, respectively, b is the width of the web, d is the effective depth of the section, d2 is the distance of the centre oiA<2 from the bottom of the section, and/cd and/yd are the design strengths of steel and concrete, respectively. In the very uncommon case where Asl <As2, the second term on the right-hand side is omitted, andv4sl is used instead of As2 in the first term.

The design value of the beam flexural capacity in positive (sagging) bending may be computed as where dx is the distance of the centre of Asi from the top of the section and bef[ is the effective width of the slab in compression.

The factor 1.3 in equation (D4.23) is meant to cover overstrength of beams, mainly due to strain hardening of steel. This value covers more than sufficiently this type of overstrength, as the reinforcing steels currently used in Europe (including its most seismic regions) are mainly of the Tempcore type, and do not exhibit large strain hardening; moreover, the overstrength of the column due to confinement of concrete is not taken into account on the left-hand side of equation (D4.23). Nonetheless, the value of 1.3 may not always be sufficient to also fully cover two other adverse effects: (1) the increased flexural capacity of the beam in negative (hogging) bending due to slab reinforcement which is parallel to the beam and is anchored in the slab within the extent of the joint or beyond (see next paragraph); and (2) plastic hinging of columns of two-way frames due to biaxiality of the bending moments.

There is ample experimental and practical evidence that, when the beam is driven past flexural yielding in negative bending and into strain hardening, such slab reinforcement up to a significant distance from the web of the beam is fully activated and contributes to the beam negative flexural capacity as tension reinforcement. Section 5 (clause 5.4.3.1.1(3)) specifies the effective in tension width of the slab on each side of the column into which the beam frames as four times the slab thickness, ht, at interior columns if a transverse beam of

Clause 5.2.3.3(2) The design value of the flexural capacity of a beam in negative (hogging) bending may be

2/7, w—> |
2 h, «-H | |

. 4h< „ | ||

2/7, k—► |
2 h, | |

-1-1 | ||

Fig. 5.3. Slab width effective as the tension flange of a beam at the support to a column, according to Section 5: (a, b) at the exterior column; (c, d) at the interior column similar size frames into the joint on the side in question, or just 2ht if there is no such transverse beam. At the two exterior columns within the plane of the frame where equation (D4.23) is checked, the above effective in-tension slab widths on each side of the web are reduced by 2hf. These slab widths, shown in Fig. 5.3, are specified in Section 5 for the dimensioning of beams at the supports to columns against the negative (hogging) bending moment from the analysis for the seismic design situation: any slab bars which are parallel to the beam and are well anchored within the extent of the joint or beyond may count as top beam reinforcement, and reduce the amount of tension reinforcement that needs to be placed within the width of the web. In that context, the value of the effective in-tension width of the slab on each side of the web has been chosen to be lower than the values of about one-quarter of the beam span suggested by practical and experimental evidence, so that it is conservative (safe-sided) for the dimensioning of beam top bars. However, it leads to underestimation of MRd b for negative bending, and hence it is on the unconservative (unsafe) side regarding prevention of column hinging through fulfilment of equation (D4.23).

5.6.2.2. The right-hand side of equation (D4.23)

The flexural capacity of a column depends on its cross-sectional shape and the arrangement of the reinforcement in it. The most common case is that of a rectangular section, with depth h (parallel to the plane within which equation (D4.23) is checked), width b, tension and compression reinforcement with cross-sectional area Asl and As2, each concentrated at a distance d1 from the nearest extreme fibres of the section in the direction of h, and additional reinforcement with cross-sectional area Asv approximately uniformly distributed along the length (h - 2d,) of the depth h between the tension and the compression reinforcement. Most often the cross-section is symmetrically reinforced: Asl =As2. However, the more general case of unsymmetric reinforcement is considered here, as it may apply also to cross-sections consisting of more than one rectangular part in two orthogonal directions, as in L-, T- or U-shaped sections. For such a section, it is most convenient to compute MRd c with respect to centroidal axes parallel to these two orthogonal directions, irrespective of the fact that they may not be principal directions. Normally - and very conveniently - the beams connected to such columns are parallel to the sides of the rectangular parts of their section, defining the framing planes within which equation (D4.23) is checked. The procedure given below for the calculation of MRd c may be applied to such sections, provided that the width of the compression zone is constant between the neutral axis and the extreme compression fibres (i.e. the depth x of the compression zone is within a single one of the rectangular parts of the section). Then, the section may be considered for the present purposes as rectangular, with constant width b, equal to that at the extreme compression fibres.

According to Eurocode 2, the design value of the flexural capacity of a cross-section, MRd, is considered to be attained when the extreme compression fibres reach the ultimate strain of concrete, ecu. The value of £cu for use in conjunction with the parabolic-rectangular a-e diagram of concrete of clause 3.1.7(1) of EN 1992-1-1 is denoted there as ecu2, and for the concrete classes common in European earthquake-resistant construction (i.e. up to C50/60) is given in Table 3.1 of EN 1992-1-1 as ecu2 = 0.0035. The concrete strain at ultimate strength, /c, i.e. at the peak of the parabolic part of the diagram, is denoted by ec, and its value for use in the calculation of the flexural capacity, ec2, is given in the same table as ec2 = 0.002 (for concrete up to C50/60).

As in primary seismic columns, and especially those which should satisfy equation (D4.23), the axial load in the seismic design situation is relatively low, the tension reinforcement, Asl, is expected to have yielded when the strain at the extreme compression fibres reaches the ultimate strain, ecu. For the grades of reinforcing steel common in Europe, the compression reinforcement, As2, being not far from the extreme compression fibres, will also be beyond its yield strain,/y/Es, when the strain at the extreme compression fibres reaches ecu. Under these conditions, the value of the neutral axis depth at ultimate moment, normalized to the effective depth of the section d = h - <i,of the section as £ = x/d, is equal to

(1 -ö^jv + uj^ -U2) + (1 + 61)Wv (1-^(1 -£c2/3ecu2) + 2Lüv

The value from equation (D5.3) (indexed by cu, to show ultimate condition controlled by the ultimate concrete strain, ecu) can be used as £ in the following equation for the flexural capacity of the column:

2 4e

The variables ujv =Asvfy/bdfc, v = N/bdfc and 6l = djd. If the design values fyd and/cd are used for/y and/c, and the conventional values ec2 = 0.002 and ecu2 = 0.0035 for ec and ecu, respectively, then equation (D5.4) gives the design value, MRd>c, of the flexural capacity.

For equation (D5.4) to be applicable for a cross-section consisting of more than one rectangular part in two orthogonal directions, with the width b taken as that of the section at the extreme compression fibres, the depth* = £cL of the compression zone calculated with the value of £ from equation (D5.3) should not exceed the other dimension (depth) of the rectangular part to which b belongs.

The column axial force, N, to be considered in the calculation of MRd c should be derived from the analysis for the seismic design situation and assume the most adverse value for the fulfilment of equation (D4.23) - i.e. minimum compression or maximum net tension - that is physically consistent with MRd c. The way to determine this value depends on the method of analysis (lateral force or modal response spectrum analysis) and on how the effects of the components of the seismic action are combined (cf. Section 4.9).

5.6.2.3. Exemptions from the capacity design rule for plastic hinging in columns (equation (D4.23)) It is extremely unlikely that both the top and bottom ends of a concrete wall within a storey will yield in opposite bending and develop plastic hinging, even when the wall section barely has the minimum dimensions required by Eurocode 8 (e.g. for a rectangular wall, just over 0.2 m x 0.8 m). So, in the horizontal direction of the building that has walls resisting at least

50% of the seismic base shear (wall and wall-equivalent dual systems), Eurocode 8 expects these walls to prevent the occurrence of a soft-storey mechanism, and waives the condition of satisfaction of equation (D4.23) at the joints of primary seismic columns with beams. In frame and frame-equivalent dual systems, fulfilment of equation (D4.23) is also waived:

8 at the joints of the top floor, as allowed for all frame structures according to Section 4.11.2.3

• at the joints of the ground storey in two-storey buildings, provided that in none of its columns the axial load ratio vA exceeds 0.3 in the seismic design situation (columns with such a low axial load ratio have good ductility and develop low P-A effects; so they can survive a displacement ductility demand equal to twice the displacement ductility factor, ¡jls, that corresponds to the value of q used in design, when a soft-storey mechanism develops at the ground storey) 0 in one out of four columns of plane frames with columns of similar size and hence of similar importance for the earthquake resistance (it may be chosen not to fulfil equation (D4.23) at interior columns rather than at exterior ones, as only one beam frames into exterior joints and it is easier to satisfy equation (D4.23) there).

At all column ends where equation (D4.23) is not checked by virtue of the exemptions above (including the columns of wall or wall-equivalent dual systems), the rules of Section 5 for buildings of DCH (but not for those of DCM) aim at a column ductility which is sufficient for development of a plastic hinge there. In fact, these rules provide the same degree of ductility as at the base of these columns, assuming that the global ductility demand is uniformly spread in all storeys.

5.6.2.4. Dimensioning procedure for columns to satisfy equation (D4.23)

Verification of equation (D4.23) at a beam-column joint pre-supposes that the longitudinal reinforcement at the end sections of the beams framing into the joint has already been dimensioned for the ultimate limit state (ULS) in bending on the basis of the analysis results for the seismic design situation and fully detailed to meet the minimum and maximum reinforcement requirements for the particular ductility class. It should be recalled that the seismic design situation is an abbreviation for the combination of (1) permanent loads entering with their nominal value, Gk, and imposed ('live') loads entering with their quasi-permanent (arbitrary point in time) value according to Section 4.4.1 and (2) the design seismic action, which includes separate consideration of each horizontal component with its own accidental eccentricity and combination of the two components (with the most adverse effect of their accidental eccentricity included) through either the square root of the sum of the squares rule of equation (D4.21) (which gives a positive end result), or the 100%-30% rule of equation (D4.22) with the internal action effects from both components normally taken with the same sign.

In principle, equation (D4.23) may well be checked after the vertical reinforcement crossing both column sections right above and below the joint is also dimensioned for the ULS in bending on the basis of the analysis results for the seismic design situation and detailed to meet the relevant detailing provisions for the particular ductility class. However, as fulfilment of equation (D4.23) is normally more demanding than the ULS in bending on the basis of the results of the analysis for the seismic design situation, it makes sense to defer dimensioning of the column vertical reinforcement until the stage at which equation (D4.23) is checked. At that stage, about half of the value of the left-hand side of equation (D4.23) may be assigned to the column section right above the joint, and the rest to the column section right below the joint. Then, the vertical reinforcement which is common in both of these sections may be dimensioned for these two uniaxial bending moments, considered to act together with the corresponding minimum value of the column axial force in the seismic design situation (determined as suggested in the last paragraph of Section 5.6.2.2, p. 98). Since, for a given vertical reinforcement, the flexural capacity increases with the (compressive) axial force, it makes sense to assign a little less than half of the left-hand side of equation

(D4.23) to the column section right above the joint, the most cost-effective apportioning being that which gives the same amount of vertical reinforcement in these two sections (a 45%/55% split is normally appropriate).

Clause The longitudinal reinforcement at the base section of the bottom storey of a column

5.5.3.2.2(14) (where the column is connected to the foundation) is dimensioned for the ULS in bending with axial force under the action effects from the analysis for the seismic design situation, without any capacity design considerations. Specifically for columns of DCH, where the seismic action effects are computed on the basis of a fairly high q factor value and may then have relatively low values, Section 5 requires that the longitudinal reinforcement placed at the base of the bottom storey is not less than that provided at the top of the storey. The objective of this requirement is to make sure that after the plastic hinge develops at the base of that column, the moment at the top does not increase to become (much) larger than at the bottom. Such an increase may unduly reduce the value of the shear span at the plastic hinge, Ls = M/V, in comparison with its value at yielding at the base, reducing also the plastic rotation capacity of the very crucial hinge at the base of the column. In terms of equations (D5.5) and (D5.8), the value of Ls in equation (D5.5) would be the initial one at yielding at the base - normally more than half the clear height of the column - while that in equation (D5.8), which determines the plastic rotation capacity, would be the subsequent smaller one.

Clause According to Section 5, the ULS verification of columns under the various combinations

5.4.3.2.1(2) of biaxial bending moments and axial force resulting from the analysis for the seismic design situation may be performed in a simplified - and safe-sided - way, neglecting one component of the biaxial bending moment at a time, provided that the other component is less than 70% of the corresponding uniaxial flexural resistance under the axial force of the combination. As one of the two components of the biaxial bending moment is normally much larger than the other in the combination, the simplified verification - devised to also cover the case of biaxial bending with about equal components - is quite conservative for the column vertical reinforcement. Where applied, it results in a sum of column flexural capacities above and below the joint, XMRd c, that exceeds the - maximum over all combinations included in the seismic design situation of the - sum of column moments above and below of the joint from the analysis, max Xme c, multiplied by 1/0.7. As max YM-e c is (about) equal to the corresponding maximum sum of beam moments on opposite sides of the joint, max XMe b, the simplified biaxial ULS verification gives ZMRdjC > max YMe b/0.7 = 1.43 max XMe b. Normally a substantial margin over max YMe, b is provided by the value of XMRd b that results from dimensioning of the beam sections next to the joint for each one of the beam moments ME b from the analysis for the seismic design situation, rounding up the reinforcement and detailing it to meet the minimum requirements (especially at the bottom of the beam). If that strength margin in the beams is about 10%, the simplified biaxial verification of the column moments gives a value of XMRiL c that automatically satisfies equation (D4.23). The implication is that dimensioning of the vertical reinforcement of the column for about half of the moment on the right-hand side of equation (D4.23) gives about the same end result as the simplified biaxial ULS verification of columns on the basis of the analysis for the seismic design situation (especially if column moments from the analysis are redistributed between the two sections above and below the joint, as permitted by clauses 4.4.2.2(1) and 5.4.2.1(1) of EN 1998-1). If the strength margin in the beams is more than 10% and/or the designer opts for a truly biaxial ULS verification of the column on the basis of the analysis results for the seismic design situation, this latter verification requires even less vertical reinforcement in the column than fulfilment of equation (D4.23), and therefore is redundant.

The conventional wisdom holds that capacity design of columns to satisfy equation (D4.23) complicates the design process. The arguments above lead to the opposite conclusion: straightforward dimensioning of the column vertical reinforcement to meet equation (D4.23) is less tedious than ULS verification of the columns on the basis of the analysis results for the seismic design situation, even when this is done with the simplified biaxial verification. If nothing else, it has to be done once in each horizontal direction (transverse axis of the column) in which equation (D4.23) has to be satisfied, whereas - due to the need to combine the components of the seismic action according to Section 4.9 and to account for the effects of accidental eccentricity - ULS verification of the columns on the basis of the analysis results for the seismic design situation normally involves four, but possibly 16, different combinations of moments with axial force. Last but not least, if the overstrength of beams relative to the requirements of the analysis for the seismic design situation is not large, then fulfilment of equation (D4.23) - at least with the value of 1.3 for the overstrength factor - does not over-penalize the column vertical reinforcement either. So, except for the top-storey columns, there is no real motivation to use the exemptions from equation (D4.23) allowed by Section 5 just for the sake of economy or simplification of the design process.

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