## Shear verification in the critical region of ductile walls

Similarly to beams and columns, the design value of the shear resistance of ductile walls, as controlled by the transverse reinforcement, FRd s, or by diagonal compression in the web, KRd max, is computed according to the rules of Eurocode 2 for monotonic loading, except for

Clauses

DCH walls and especially in their critical region. The special rules applicable for DCH walls are detailed below.

In the critical region of DCH walls, the design value of the cyclic shear resistance, as controlled by diagonal compression in the web, Vm max, is taken as just 40% of the value given by Eurocode 2 for monotonic loading. It was found by Biskinis et al.69 that cyclic loading drastically reduces this particular shear resistance of walls, and they fitted the following expression to it, adopted in Annex A of EN 1998-352 (with units of meganewtons and metres):

N Afc h

(l + 0.25max(1.75; 100ptot))| l-0.2min(2; -j- )^min(100; fc)bvz

The variables in equation (D5.47), including the plastic part of the chord rotation ductility factor, pf = pe - 1, are as defined for equations (D5.36)-(D5.38). The limited test results for shear failure under cyclic loading by diagonal compression in the web prior to flexural yielding suggest that equation (D5.47) holds in that case as well, with pf - 0. The test data to which equation (D5.47) was fitted show that, for values of pB representative of ductility demands in DCH walls, on average, the Eurocode 2 value of VR max (using the actual value of fc in lieu of/cd) gives 40% of the experimental cyclic shear resistance. Hence the relevant rule of Section 5 for the critical region of DCH walls. The difference is very large, and normally it should have been taken into account in the Eurocode 8 rules for the shear design of ductile walls of DCM as well. It was feared, though, that a large reduction of the design shear resistance, when applied together with the magnification of shears by the factor of equation (D5.19), might be prohibitive for the use of ductile concrete walls in earthquake-resistant buildings. So, it was decided to leave the design rules for DCM walls unaffected, at least until the reduction demonstrated by the currently available data is supported by more test results. For the time being, the designer is cautioned to avoid exhausting the present liberal limits for DCM ductile walls against diagonal compression in the web.

The second point where shear design of DCH walls deviates from the general Eurocode 2 rules is in the calculation of the web reinforcement ratios, horizontal ph and vertical in those storeys of DCH walls where the shear span ratio, as = -MEd/FEd/w, is less than 2. The maximum value of MEd in the storey (normally at its base) is used in the calculation of o:s. Significant uncertainty exists regarding the cyclic behaviour of walls with as<2 that ultimately fail by diagonal tension (and hence are controlled by the web reinforcement), as most of the walls with as < 2 which have been cyclically tested in the laboratory have failed by diagonal compression (and hence are included in the data that support equation (D5.47)). Unlike the relative abundance of data on this latter type of wall, only four out of the 26 laboratory walls that failed in shear by diagonal tension after flexural yielding and support equations (D5.36)-(D5.38) have as < 2. In view of the lack of information specific to cyclic loading, the following modification of the rule given in clause 6.2.3(8) of Eurocode 2 for the calculation of the transverse reinforcement in members with 0.5 < as<2 under monotonic loading has also been adopted for the determination of ph in those storeys where a, < 2:

Vr fy yh,d

Ed y where ph is the ratio of horizontal reinforcement, normalized to the thickness of the web, bwo, and /yh d is its design yield strength. A Vc term has been included, equal to the design shear resistance of concrete members without shear reinforcement according to Eurocode 2, FRd c. If bwo and the effective depth, d, of the wall are expressed in metres, the wall gross cross-sectional area, y4c, in square metres, KRd c and the wall axial force in the seismic design situation, NEA, in kilonewtons and if/ck is in megapascals, KRd c, as given in Eurocode 2, is

where p, denotes the tensile reinforcement ratio, and -jc is the partial factor for concrete. However, in the critical region of walls, KRdjC = 0 if NEd is tensile (negative). The ratio of vertical web reinforcement, pv, is then dimensioned to provide a 45° inclination of the concrete compression field in the web, together with the horizontal reinforcement and the vertical compression in the web due to minimum axial force in the seismic design situation, min NEd. There is certainly room for future improvement of these rules, once more data become available on the cyclic behaviour and failure of low-shear-span-ratio walls by diagonal tension.

As mentioned in the closure to Section 5.7.6, sections of a DCH wall within its critical Clause region should be verified against sliding shear. Verification may be limited to the storey end 5.5.3.4.4( I) section(s) within the wall critical region, normally coinciding with a construction joint. If the critical region of the wall is limited to its bottom storey, only the base section needs to be verified.

The design resistance against sliding shear comprises three components: Clause

(1) A dowel action term, equal to the minimum of the following:

- The resistance of vertical bars in pure shear, taken as 0.25L4sv/yd, where/lsv is the total area of the vertical bars in the web plus any additional vertical bars placed in the boundary elements specifically for the purpose of resistance to shear sliding without counting in the flexural reinforcement. The safety factor with respect to the yield force of a bar in pure shear (i.e. without axial force), which is equal toAsvfydH3, has a value of 2.3.

The dowel action resistance, as determined by the interaction between the bar and the surrounding concrete, taken to be equal to 1.3/1 sv(/yd/:d)1/2 with As as defined above. The safety factor with respect to the monotonic dowel action resistance of stress-free bars deeply embedded in concrete, which is equal to l3dbL2(fydfcd)112, is then 4/tt = 1.275.

For lower concrete classes, e.g. below C25/30, the term 0.25/4sv/yd governs. For the contribution of a bar to these sources of resistance to be fully available, its concrete cover should be at least 3dbL in the direction of the thickness of the wall, at least 8<ibL along the wall length ahead of the bar (i.e. towards the compression zone of the section) and at least 5dhL behind it. These fairly restrictive conditions and the reduction of dowel action resistance with the axial stress level in the bar are behind the large hidden safety factors mentioned above and the exclusion from Asv of those vertical bars in the boundary elements that count as flexural reinforcement.

(2) The contribution of the compression zone, taken to be equal to the minimum of the following:

- The shear resistance as controlled by diagonal compression over the compression zone, computed as if the latter were a beam of rectangular section with effective depth that of the compression zone, x, and thickness that of the web, b,m. This calculation employs an inclination 6 of the compression struts equal to 45° and the reduction factor 0.6(1 -/ck(MPa)/250) on /cd (the factor v of clause 6.2.3 in Eurocode 2, or t] of Section 5.7.5 and equations (D5.31) and (D5.32) above).

- The frictional resistance, taken to be equal to the friction coefficient /../. multiplied by the normal force on the compression zone. This latter force is taken to be equal to the compression force, MEd/z, delivered to the compression zone by the bending moment from the analysis in the seismic design situation, MEd, plus the share of the compression zone to the total clamping force developing over the cross-section at imminent sliding, Asyfyd + NEd. Considering this force as uniformly distributed along the length of the wall, /w, the share of the compression zone is equal to its depth, x, normalized to /w. The values provided for /_/. in Eurocode 2 may be used, with H = 0.6 - applicable to smooth interfaces - being more appropriate at construction joints and ¡i = 0.7 - for rough ones - at cracks that may develop during the response in monolithic concrete. Normally the former term (that due to diagonal compression) governs. Clauses (3) The horizontal components AJyd sin a of the yield force - in tension and compression -

5.5.3.4.4(4), of bars placed, at an angle ±a to the vertical and with cross-sectional area As per

5.5.3.4.4(5) direction, specifically to resist sliding shear. It is recommended that inclined bars are placed so that they cross the base section of the wall at its mid-length, to avoid affecting -through the couple of the vertical components of their tension and compression forces -neither its flexural capacity, MRdo, used for the calculation of the design shear, VEd, according to equations (D5.17) and (D5.18), nor the location of the plastic hinge. A value of the inclination a - 45° is not only convenient but also the most cost-effective, in view of the requirement of Section 5 that inclined bars extend up to a distance of at least 0.5/w above the base section.

Clause Inclined bars should normally be placed only if the two other components of the resistance

5.5.3.4.4(3) against sliding shear (listed under 1 and 2 above) are not sufficient. However, Section 5 requires that they are always placed at the base of squat DCH walls - i.e. of those with a height-to-length ratio less than 2 - in a quantity sufficient to resist at least 50% of the design shear there, VEd; moreover, in such walls inclined bars are required at the base of all storeys in a quantity sufficient to resist at least 25% of the storey design shear.

Clause

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