Simple rules for the estimation of the flexure- or shear-controlled cyclic ultimate

3.1.1.1 Flexure-controlled ultimate chord rotation under cyclic loading Ultimate curvature and ultimate chord rotation from the plastic hinge length

Lacking experimental measurements of curvatures at flexure-controlled ultimate conditions in circular RC piers, it is presumed that the material parameters (for confined or unconfined concrete and for the available elongation of tension reinforcement) fitted in Section 3.1.3.2 of Part I to measured ultimate curvature, φu, of RC members with rectangular compression zone (typical of building construction) and to the associated fixed-end rotation, all apply to circular RC piers as well. On the basis of these material parameters, the ultimate curvature, φu, of circular piers is determined on the basis of the plane sections hypothesis, supplemented with the following material σ-ε laws:

- an elastic-linearly-strain hardening σ-ε law for steel;

- a parabolic σ-ε diagram for concrete up to the compressive strength fc at a strain εco=0.002 (or fcc for confined concrete), followed by a horizontal branch up to εcu,c, given by Eq. (3.22) in Section 3.1.3.2 of Part I for confined concrete.

The neutral axis depth satisfying force equilibrium over the section can be determined only iteratively. Eqs. (3.1), (3.2) in Section 3.1.3.1 of Part I are used to obtain the ultimate curvature, φu, from the neutral axis depth.

For hollow rectangular piers, Sections 3.1.31 and 3.1.3.2 of Part I apply fully. If the neutral axis depth, ξud, computed on the basis of these sections exceeds the thickness of the compression flange, an iterative algorithm is used, with the same basis as that employed for circular piers, to take into account the U shape of the compression zone.

The ultimate chord rotation, θu, of hollow rectangular piers can be determined on the

basis of Section 3.1.3.3 of Part I, using Eq. (3.26b) therein for the plastic hinge length under cyclic loading and Eq. (2.7b) of Section 2.2.1.3 in the present Part II for the chord rotation at yielding, θy. For circular piers, the fitting to the available results of cyclic tests to flexure-controlled failure gave the following result, to be used in Eq. (3.25) in Section 3.1.3.3 of Part I, together with the outcome of Section 2.2.1.2 of the present Part II for the yield curvature, φy, and of Eq. (2.7a) in Section 2.2.1.3 for θy:

Circular piers - cyclic loading: _pl 2 0.09 ^s 3

L = L + D (3.1)

Table 3.1 gives statistics of the ratio of the experimental ultimate chord rotation to the one predicted as outlined above, for piers failing in flexure under cyclic loading.

Empirical ultimate chord rotation of hollow rectangular piers

As the fitting to the test results on flexure-controlled ultimate chord rotation of hollow rectangular piers by the model above and Eq. (3.26b) in Sections 3.1.3.3 of Part II, although optimal, is not very satisfactory, the empirical alternatives of Sections 3.1.3.4 of Part I, Eqs. (3.27), were extended to this type of RC member. This entails:

- taking for θu the outcome of Eq. (3.27a) in Section 3.1.3.4 of Part I, times 5/6, or - taking for θu the sum of the outcome of Eq. (2.7a) in Section 2.2.1.3 for θy, plus that

of Eq. (3.27a) in Section 3.1.3.4 of Part I, times 0.77.

Rows 3 and 4 in Table 3.1 give statistics of the ratio of the experimental ultimate chord rotation to the so-predicted one, for piers failing in flexure under cyclic loading.

Another purely empirical alternative for the plastic component of the ultimate chord rotation, θupl =θu-θy, can cover all type of RC members with rectangular compression zone, without distinction between walls, hollow rectangular piers or beam/columns. It is given by the following expression, with the elastic component, θy, given by Eq. (2.7a) in Section 2.2.1.3 for hollow rectangular piers and by Eq.(2.11) in Section 2.2.1.3 of Part I for other types of members:

( )

( ) ( ) max(0.01, ') ^{1/3 0.2 100 100}

1 0.525 1 0.6 1 0.05max 1.5,min 10, 0.2 25 1.225 max(0.01, )

yw sx

c d

pl u

f s f

st cy sl c

w

h L

a a a f

b h

ν αρ ρ

θ

ω ω

⎛ ⎞

⎜ ⎟

⎜ ⎟

⎝ ⎠

=

⎛ ⎛ ⎛ ⎞⎞⎞ ⎛ ⎞

− + ⎜⎜⎝ − ⎜⎜⎝ ⎜⎝ ⎟⎠⎟⎟⎠⎟⎟⎠ ⎜⎝ ⎟⎠

(3.2) where (cf. Section 3.1.3.4 of Part I):

ast: coefficient for the type of steel, equal to ast = 0.022 for ductile hot-rolled or for heat-treated (tempcore) steel and to ast =0.0095 for cold-worked steel;

acy: zero-one variable, equal to 0 for monotonic loading and to 1 for cyclic loading;

asl: zero-one variable for slip, equal to 1 if there can be slip of the longitudinal bars

from their anchorage beyond the section of maximum moment, or to 0 if not;

bw/h ratio of thickness of one web to the section depth;

ν=N/bhfc (with b=width of compression zone, N=axial force, positive for compression);

ω1: mechanical reinforcement ratio of tension and “web” longitudinal reinforcement, (ρ1 fy1+ρvfyv)/fc;

ω2: mechanical reinforcement ratio of compression longitudinal reinforcement, ρ2fy2/fc; fc: uniaxial (cylindrical) concrete strength (MPa)

Ls/h=M/Vh: shear span ratio at the section of maximum moment;

ρs=Ash/bwsh: ratio of transverse steel parallel to the direction of loading;

fyw: yield stress of transverse steel;

α_: confinement effectiveness factor from Eq. (3.21) in Section 3.1.3.4 of Part I;

ρd: steel ratio of diagonal reinforcement in each diagonal direction.

Table 3.1, rows 5-11, gives statistics of the ratio of the experimental ultimate chord rotation to the one predicted as outlined above, for all types of RC members with rectangular compression zone that fail in flexure. The statistics for hollow rectangular piers under cyclic loading are slightly inferior to the other empirical alternatives as far as the median is concerned, but better for the scatter. The overall agreement of Eq. (3.2) to the experimental results for all types of members is almost as good as that of the Eqs.

(3.27) in Section 3.1.3.4 of Part I (cf. Table 3.3 therein, rows 7 to 18).

Table 3.1: Mean, median* and coefficient of variation of ratio of experimental-to-predicted ultimate chord rotation.

Ultimate chord rotation predicted by different models for different types of members

No.

testsmean* median* coef. of variation θu,exp/θu, Eq.(3.25)-PartI & Eq.(3.1)-Part II – Circular piers, cyclic loading 110 1.02 1.005 29.2%

θu,exp/θu,Eq.(3.25),(3.26b)-Part I - hollow rectangular piers, cyclic loading 30 0.98 0.99 42.0%

θu,exp/θu, (5/6)xEq.(3.27a)-Part I - hollow rectangular piers, cyclic loading 30 0.955 1.015 34.4%

θu,exp/θu, 0.77xEq.(3.27b)-Part I - hollow rectangular piers, cyclic loading 30 0.94 1.005 33.3%

θu,exp/θu, Eq. (3.2) – hollow rectangular piers, cyclic loading 30 1.025 1.05 29.5%

θu,exp/θu, Eq. (3.2) – All tests 1307 1.065 0.995 42.9%

θu,exp/θu, Eq. (3.2) – All monotonic tests 295 1.15 1.015 53.0%

θu,exp/θu, Eq. (3.2) – All cyclic tests 1012 1.04 0.995 38.3%

θu,exp/θu, Eq. (3.2) – w/o slippage of bars from anchorage 211 1.145 0.995 50.3%

θu,exp/θu, Eq. (3.2) – w/ slippage of bars from anchorage 1096 1.05 0.995 40.8%

θu,exp/θu, Eq. (3.2) – Walls w/ slippage of bars from anchorage 78 1.05 0.99 33.4%

*^*See footnote of Table 2.1 for the median vs the mean for large sample size.

3.1.1.2 Shear resistance in diagonal tension or compression under inelastic cyclic deformations after flexural yielding

The models presented in Section 3.1.6 of Part I for the shear resistance in diagonal tension under inelastic cyclic deformations after flexural yielding, and in Section 3.1.7 of Part I for the cyclic shear resistance of walls or squat columns in diagonal compression, have been fitted to databases that include shear-critical circular and hollow rectangular piers. So, they are applicable to these types of RC members, as well.

For circular piers, in particular, the following apply:

- In Eqs. (3.29), (3.31)-(3.33), the depth of the cross-section, h, is the pier diameter D, - The cross-section area, Ac, in Eq. (3.29) is taken equal to πDc2/4, where Dc is the

diameter of the concrete core centreline the circular hoops,

- In Eq. (3.29) the contribution of transverse reinforcement to shear resistance, Vw, is:

) 2 2 (

π f D c

s

V A _yw

h

w= sw − (3.3)

where Asw is the cross-sectional area of a circular stirrup, fyw is its yield stress, sh is the centreline spacing of stirrups and c the concrete cover to reinforcement.

Table 3.2: Mean*, median* & coef. of variation of ratio of experimental-to-predicted shear resistance

Shear resistance for different failure modes # of

data mean*median

*

coef. of variation VR,exp/VR,Eq.(3.29a)-Part I - circular piers, diagonal tension failure 68 1.055 1.03 16.2%

VR,exp/VR,Eq.(3.29a)-Part I - hollow rect. piers, diagonal tension failure 24 1.03 0.98 17.4%

VR,exp/VR,Eq.(3.29b)-Part I - circular piers, diagonal tension failure 68 1.02 1.015 15.9%

VR,exp/VR,Eq.(3.29b)-Part I - hollow rect. piers, diagonal tension failure 24 1.09 1.05 16.4%

VR,exp/VR,Eq. (3.31)-Part I - hollow rect. piers, web compression failure 51 1.035 1.01 17.2%

*See footnote of Table 2.1 for the median vs the mean for large sample size.

Table 3.2 gives statistics of the ratio of the experimental the experimental to the so-predicted shear resistance for shear-critical circular or hollow rectangular piers. Note that

Eqs. (3.29), (3.31), (3.32) have been fitted to all types of RC members, regardless of cross-sectional shape. So, the deviations of the median from 1.00 in Table 3.2 should be viewed in conjunction with those in its counterpart in Part I (Table 3.5 in Section 3.1.6 therein).

3.1.2 Models/procedures for estimation of ultimate deformations and shear

In document Guidlines for Displacement-Based Design of Buildings and Bridges (Page 158-162)