fardis paper otani

4-3: Fardis Otani Symposium 2003

DEFORMATION CAPACITY OF RC MEMBERS, AS CONTROLLED BY FLEXURE OR SHEAR

Michael N. Fardis and Dionysis E. Biskinis

Structures Laboratory, Department of Civil Engineering, University of Patras, Greece E-mail: [email protected]

SUMMARY A large database of monotonic and cyclic uniaxial test results is utilised to develop models for the deformation capacity of R/C members. The database covers beams, columns with rectangular or circular section, walls with rectangular, T- or barbelled section, and hollow rectangular piers. Most of the tests in the database are up to specimen to failure, conventionally defined as a post-peak drop in lateral load resistance by at least 20%. One type of models developed for the chord rotation (or drift) capacity for flexure-controlled failure is based on curvatures in the plastic hinge - at yielding and ultimate, calculated on the basis of first principles and of different confinement models at ultimate - and on expressions for the plastic hinge length empirically fitted to the data. Except for columns with circular section, the predictions of this type of models are characterised by unacceptably large scatter and, often, by significant bias. Purely empirical models, statistically fitted to the data, are found to offer better predictive capability for the flexure-controlled chord rotation capacity of all types of members in the database with rectangular or quasi-rectangular section. For members under cyclic loading ultimately failing in shear after yielding in flexure, expressions of the familiar type are developed for the reduction of shear resistance with the chord rotation ductility ratio. These expressions, applicable over all types of members in the database, are characterised by low scatter for the prediction of shear resistance in terms of the post-elastic cyclic displacements, but cannot be meaningfully inverted to give a shear-controlled ductile deformation capacity. A model based on first principles with empirical corrections is also developed for the chord rotation (drift) at member yielding, as a tool for the models of deformation capacity as controlled by flexure or shear, as well as for the calculation of the secant stiffness of members at yielding.

1. INTRODUCTION Recent years have seen a large interest of the international earthquake engineering community in the quantification of the deformation capacity of R/C members. The emergence of procedures for seismic assessment of existing structures which entail member verifications explicitly in terms of deformations [ASCE, 2000 and 2001, Comité Européen de Normalisation, 2003a] and the forthcoming codification of the design of new structures directly on the basis of nonlinear analysis with explicit checks of member deformations [Comité Européen de Normalisation, 2003b], provide strong motivation for this interest. To support the effort of quantification of the deformation capacity of R/C members, the senior author of the paper and his co-workers have been assembling at the University of Patras over a period of almost ten years a databank of tests on R/C members. Early models for the deformations of R/C members at yielding and at flexure-controlled failure under monotonic or cyclic loading have been developed on the basis of the databank and reported in [Panagiotakos and Fardis, 2001]. The results of more recent efforts, still on flexure-controlled members, are included in [Federation Internationale du Beton, 2003a, 2003b]. The present paper represents a major stride in this on-going effort, not only because the database has recently been re-evaluated and increased in size by almost 50%, but mainly because it has been used also for the quantification of the resistance and deformation capacity of members which are controlled by shear.

511

Table 1 Database of member tests with measurement of deflection of shear span with respect to member axis at fixed end (chord rotation).

slip of long. bars from anchorage

Load history and mode of failure

monotonic beyond flexural yielding cyclic beyond flexural yielding cyclic beyond

shear yielding flexural failures shear failures yes no all no

failureflexural failures

all no failureconforming

detailing non-conform.

detailing: all web tension

web com-pression sliding

failure by web tension

failure by web com-pression

All tests beyond flexural yielding(16)=(6)+

(7)+ (10)+(11)

+(12)+(13)

All flexural failures

(17)= (5)+(10)

member type and cross-section

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (a) Column - Circular 160 0 160 3 0 3 39 76 0 76 44 0 0 19 0 162 76

(b) Column - Rectangular, conventional reinf/ment 1080 (ρ=ρ΄, ρv≥0, ρd=0, ν≥0)

77 1157 4 56 60 102 727 33 760 146 0 0 47 0 1068 816

(c) Column - Rectangular, diagonally reinforced (ρ=ρ΄, ρv≥0, ρd>0, ν≥0)

59 0 59 0 4 4 3 52 0 52 0 0 0 0 0 59 56

(d) Beam - Rectangular or T 132 149 281 0 215 215 0 52 1 53 0 0 0 0 0 268 268 (e)=(b)+(c)+(d)

All beams or columns w/ rectang. web 1271 226 1497 4 275 279 105 831 34 865 146 0 0 47 0 1395 1140

(f) Wall - Rectangular 63 0 63 0 1 1 9 46 2 48 1 1 5 0 0 65 49 (g) Wall - T or barbelled 46 0 46 2 3 5 3 9 0 9 5 26 2 0 10 50 12

(h)= (f)+(g) Wall - All 109 0 109 2 4 6 12 57 0 57 6 27 7 0 10 115 61

(i)= (e)+(h)

All columns, beams or walls w/ rectang. web 1380 226 1606 6 279 285 117 888 34 922 152 27 7 47 10 1510 1201

(j) Pier - hollow rectangular 36 0 36 0 0 0 8 23 3 26 11 0 0 7 0 45 26 (k)=

(i)+(j) All members with rectangular web(s) 1416 226 1642 6 279 285 125 911 37 948 163 27 7 54 10 1555 1227

(l) All tests 1576 226 1802 9 279 288 164 987 37 1024 207 27 7 73 10 1717 1303


2. THE DATABASE OF TEST RESULTS ON R/C MEMBERS The databank of tests used in this paper comprises mainly specimens subjected to uniaxial transverse (i.e. lateral) loading with or without axial load - constant or varying. Of interest here are only specimens of that type. The main reason for assembling the database has been the development of models for the deformations that develop over the shear span, Ls, of R/C members fixed at the section of maximum moment. Therefore, the databank is limited mainly to tests which report the transverse deflection at, or near, the point of zero-moment with respect to the specimen axis at the section of maximum moment. Such tests are on simple- or double-cantilever specimens, or on simply-supported beams loaded only at mid-span. This deflection is used here in the form of drift (ratio or angle), θ, i.e. deflection divided by the distance from the point of measurement to the section of maximum moment. Normally this distance is equal to the shear span or a multiple of it, and then θ is also the chord rotation at the section of maximum moment. Table 1 gives the breakdown of this type of specimens in the databank, depending on the specimen geometry (beams, columns - with conventional or diagonal reinforcement -, walls, or piers), type of loading (monotonic or cyclic), the mode of yielding or failure (due to flexure or shear, by web crushing or interface shear, etc.), and the occurrence or not of (bond-) slippage of longitudinal bars from their anchorage zone beyond the section of maximum moment. Normally, in simple- or double-cantilever specimens a certain amount of such slippage (pull-out) takes place, producing a fixed-end rotation that contributes to the deflection of the shear span. Due to symmetry, there is no such slippage in simply-supported beam specimens loaded only at mid-span, except when the load is applied through a bulky stub, long enough for rebar slippage to develop on both sides of the mid-span.

Table 2 Database of member tests with measurements of curvatures

load history and mode of failure cyclic beyond flexural yielding

slip of longitudinal.

bars from anchorage

flexural failures

yes no

All

(3)= (1)+(2)

monotonic beyond flexural

yielding -all flexural

failure

non- failure conforming

detailing non-conforming

detailing

All

(8)= (6)+(7)

All tests beyond flexural yielding

(9)=

(4)+(5) +(8)

All flexuralfailures

(10)= (4)+(8)

member type and cross-section

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

(a) Column - Rectang-ular, conventional-ly reinforced

138 36 174 49 1 70 6 76 126 125

(b) Beam - Rectangular or T 37 162 199 181 0 6 0 6 187 187

(c)= (a)+(b)

All beams orcolumns withrectangular web

175198 373 230 1 76 6 82 313 312

(d) Wall - Rectangular 5 0 5 0 0 3 2 5 5 5

(e) Wall - T- or barbelled 7 0 7 0 0 1 0 1 1 1

(f)= (d)+(e) Walls - All 12 0 12 0 0 4 2 6 6 6

(g)= (c)+(f)

All columns,beams and walls 187198 385 230 1 80 8 88 319 318

(h) Pier - Hollow rectangular 14 0 14 0 0 8 0 8 8 8

(j)= (g)+(h) All members 201198 399 230 1 88 8 96 327 326

In some tests of the databank the relative rotation between the section of maximum moment and a nearby section within the plastic hinge region has been measured and translated into an average curvature, φ, of the plastic hinge region, including or not effects of reinforcement slippage from its anchorage beyond the

513


514

section of maximum moment (Table 2). Some of these tests are not included in Table 1, because deflections with respect to the specimen axis at the section of maximum moment were either not reported, or could not be derived from the measurements (e.g. in beams with loading other than at mid-span alone). Most of the tests in the databank were continued up to specimen ultimate conditions (failure), identified with a distinct change in the measured lateral force-deformation response: in monotonic loading, with a noticeable drop of lateral force after the peak (at least 20% of the maximum resistance); in cycling loading, with a distinct reduction of the reloading slope, or of the area of the hysteresis loops, or of the peak force, compared to those of the preceding cycle(s), typically associated with a drop in resisting force greater than 20% of the maximum resistance. Reinforcing steel in the tests has been classified into three grades. Most specimens have ductile hot-rolled steel with hardening ratio, ft/fy, in the order of 1.5 and strain at peak stress, εsu, around 15%. In European tests heat-treated tempcore steel has been used after the early ‘90s, with a value of ft/fy around 1.2 and of εsu of the order of 8%. In about 60 monotonically tested European specimens, brittle cold-worked steel has been used, with a value of ft/fy about 1.1 and of εsu around 4%. Table 3 Mean*, median* and coef. of variation of ratio experimental-to-predicted values at yielding

Quantity # of data mean* median* coefficient

of variation θy,exp/θy,Eq.(2) Beams, columns w/ rectangular section - w/o slip 198 1.05 1.00 29.5% θy,exp/θy,Eq.(2) Beams, columns w/ rectangular section - w/ slip 1151 1.05 0.995 40.0% θy,exp/θy,Eq.(2) Beams, columns w/ rectangular section - All 1349 1.05 0.995 38.6% θy,exp/θy,Eq.(3) Walls (all w/ slip) 145 1.015 0.99 32.5% θy,exp/θy,Eq.(4) Columns w/ circular section (all w/ slip) 160 1.05 0.99 33.4% ϕy,exp/ϕy,pred.-1st-principles Beams, columns w/ rect. section - w/o slip 198 1.325 1.275 29.3% ϕy,exp/ϕy,pred.-1st-principles Beams, columns w/ rect. section - w/ slip 175 1.205 1.06 37.6% ϕy,exp/ϕy,pred.-1st-principles Beams, columns w/ rect. section - All 373 1.27 1.205 33.4% My,exp/My,pred.-1st-principles Beams, columns & walls w/ rectangular section 1513 1.025 1.015 16.2% My,exp/My,pred.-1st-principles Columns w/ circular section 181 1.015 1.005 16.7% (MyLs/3θy)exp/(MyLs/3θy)pred Beams, columns & walls w/ rectan. section 1412 1.10 1.035 40.9% (MyLs/3θy)exp/(MyLs/3θy)pred Columns w/ circular section 152 1.07 1.035 31.2% *If the coefficient of variation is high, the median is more representative of the average trend than the mean, as the median of the ratio predicted-to-experimental value is always the inverse of the median of the ratio experimental-to-predicted value, whereas the mean of both ratios is typically greater than the median. 3. DRIFT (OR CHORD ROTATION) AT MEMBER YIELDING The value of the chord rotation at yielding at the corresponding end of the member, θy, is important as the baseline for the plastic component of the ultimate chord rotation (plastic rotation capacity), as well as the normalizing factor of chord rotation (total or plastic component), whenever this latter is expressed as ductility ratio, µθ. More importantly, θy determines, through Eq.(1), the value of the secant stiffness of the shear span, Ls, at member yielding, often taken as the effective elastic stiffness in a bilinear force-deformation model of the shear span under monotonic loading:

y

syeff

LMEI

θ3= (1)

where My is the yield moment in the bilinear M-θ model of the shear span. The following expressions were derived from those tests in the databank with yielding in flexure: For beams or columns with rectangular section:


c

ybysl

syy f

fddd

aL 2.0)'(

00275.03 −

++=ε

φθ (2)

For walls with rectangular, T-shaped, or H-shaped section, or for barbelled walls:

c

ybysl

syy f

fddd

aL 2.0)'(

0025.03 −

++=ε

φθ (3)

For columns with circular cross-section:

c

ybysl

syy f

fdaL 2.0

002.03

φφθ ++= (4)

If φy denotes the yield curvature at the section of maximum moment and if the moment diagram is linear over the shear span, the 1st term in Eqs.(2)-(4) is the contribution of flexural deformations to θy. The 2nd term represents the magnitude of shear deformations within the shear span at flexural yielding and has been found to be practically independent of any factor other than the type of member in Eqs.(2)-(4). The 3rd term is the fixed-end rotation due to bar pull-out from the anchorage zone and does not appear in Eqs. (2)-(4) when such pull-out is not physically possible. In that case asl=0 is used in Eqs. (2)-(4) for the zero-one variable asl, whereas asl=1 applies if such pull-out is considered as possible. In the 3rd term, εy is the yield strain of the tension reinforcement, d or d’ denote the effective depth to the tension or to the compression reinforcement, respectively, db is the diameter of the tension reinforcement and fy, fc the yield stress of tension reinforcement and the compressive strength of concrete (both in MPa). The 3rd term corresponds to a linear reduction of steel stress from fy to zero over a development length deriving from a mean bond stress of 0.625 cf (which is low in comparison to the ultimate bond stress of about 2 cf or

2.5 cf in unconfined or confined concrete respectively, but gives the best fit of Eqs.(2)-(4) to the data).

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3 3

Θy,pred (%)

Θy,

exp

(%)

.5

median: θy,exp=θy,pred

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4Θy,pred (%)

median: θy,exp=θy,pred

0

0.4

0.8

1.2

1.6

2

2.4

0 0.4 0.8 1.2 1.6 2 2.4

Θy,pred (%)

median: θy,exp=0.99θy,pred

(a) beams or columns; rectangular section, Eq.(2)

(b) walls; rectangular, T- or H-section, Eq.(3)

(c) columns; circular section, Eq.(4)

Figure 1 Comparison of experimental chord rotations at yielding to values from Eqs.(2)-(4) The fit of Eqs.(2)-(4) to the data is shown in Figure 1, while the statistics of the ratio of experimental to predicted values are given in Table 3. This fit corresponds to a yield curvature, φy, derived from first principles. More specifically, for rectangular or T-sections, φy is derived on the basis of the plane-section hypothesis, of equilibrium, and of linear σ-ε laws up to a steel strain of εy if section yielding is controlled by the tension reinforcement, or up to a concrete strain of 0.9fc/Ec if it is controlled by the concrete in compression (see [Panagiotakos and Fardis, 2001] for expressions for φy). Similar assumptions are employed for circular sections, except that the concrete σ-ε law is taken parabolic up to a strain of 0.002 and recourse has to be made to an iterative algorithm [Biskinis et al, 2002]. Possibly due to the way in which experimental curvatures are derived from the relative rotations of two sections within the plastic hinge, the expressions developed for φy from first principles do not provide unbiased estimates of the yield curvature in the 373 tests where such curvatures were measured. This is clear from Figure 2 and from the fact that median ratios of experimental-to-predicted φy, listed in Table 3,

515


are well above 1.0 (interestingly, they are greater in the 198 tests in which measured curvatures were not affected by slippage of the rebars from the anchorage zone, than in the 175 tests where they were affected). Despite this discrepancy between measured values of φy and the ones calculated from first principles, a closer fit to the data is possible through expressions of the type Eqs.(2)-(4) if these latter values are used instead of the empirical expressions: φy = 2.1εy/d or φy = 1.9εy/h which were found to provide an unbiased fit to the measured values of φy in the 373 tests, without a significantly increased scatter over that of the fundamental expressions for φy.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.01 0.02 0.03 0.04 0.05 0.06

φy,pred (1/m)

φy,

exp

(1/m

)

median: φy,exp=1.28φy,pred

0

0.01

0.02

0.03

0.04

0.05

0 0.01 0.02 0.03 0.04 0.05

φy,pred (1/m)

walls

median: φy,exp=1.06φy,pred

(a) beams and columns with rectangular section;

no slip of longitudinal bars (b) beams and columns with rectangular section;

slip of longitudinal bars Figure 2 Comparison of experimental curvatures at yielding to values calculated from 1st principles

0

2000

4000

6000

8000

10000

0 2000 4000 6000 8000 10000

My,pred (kNm)

My,

exp

(kNm

)

rectangular

w alls&hollow

median: My,exp=1.02My,pred

0

200

400

600

800

1000

1200

0 200 400 600 800 1000 1200

My,pred (kNm)

My,

exp

(kN

m)

median: My,exp=My,pred

(a) rectangular sections (b) columns with circular section

Figure 3 Comparison of experimental yield moments to values calculated from 1st principles To complete the picture regarding the application of Eq.(1) to estimate the secant stiffness of the shear span, Ls, at member yielding, the yield moment My calculated from first principles (with same assumptions and references mentioned above for the calculation of the yield curvature, φy) is compared in Figure 3 to the experimental “yield moment”, taken as the moment at the corner of a bilinear M-θ curve fitted to the envelope of the measured M-θ response of the shear span. Statistics of the ratio of the experimental yield moment to the calculated value are listed in Table 3. The median value of this ratio exceeds 1.0, as the corner of the bilinear approximation to the experimental M-θ curve lags behind first

516


yielding of the tension reinforcement or strong nonlinearity of the extreme compression fibers. The difference is greater in circular sections, as there the experimental M-θ response curves down more gradually. So, for such sections the experimental value of My is compared in Figure 3 and in Table 3 to the average of My and of the theoretical ultimate moment, Mu, of the section, both computed (through iterations) on the basis of first principles as described in [Biskinis et al, 2002]. Statistics of the ratio of the experimental secant stiffness at member yielding to the value calculated from Eq.(1) on the basis of the yield moment My from first principles and the values of θy from Eqs.(2)-(4) are also listed in Table 3. For members with rectangular or circular section, the experimental secant stiffness at member yielding is on average 25% or 30%, respectively, of that of the uncracked gross section; but its scatter about this latter values is far greater (around 70%) than about the value given by Eq.(1). Table 4 Mean, median, coefficient of variation of ratio experimental-to-predicted values at ultimate

Quantity # of data mean median coefficient

of variationϕu,exp/ϕu,pred.-1st-principles-Mander-Eq.(6) Monotonic loading - Rect. sections 230 1.12 0.995 64.1% ϕu,exp/ϕu,pred.-1st-principles-Priestley-Eq.(7) Monotonic loading - Rect. sections 230 1.055 0.715 97.7% ϕu,exp/ϕu,pred.-1st-principles-CEB/FIP-Eq.(8) Monotonic loading - Rect. sections 230 2.36 1.97 69% ϕu,exp/ϕu,pred.-1st-principles-proposed Eq.(11) Monotonic loading - Rect. sections 230 1.09 0.995 56.5% ϕu,exp/ϕu,pred.-1st-principles-Mander-Eq.(6) Cyclic loading - Rectangular sections 89 1.35 1.13 77.5% ϕu,exp/ϕu,pred.-1st-principles-Priestley-Eq.(7) Cyclic loading - Rectang. sections 89 0.955 0.545 130.5%ϕu,exp/ϕu,pred.-1st-principles-CEB/FIP-Eq.(8) Cyclic loading - Rectang. sections 89 1.905 1.53 66.8% ϕu,exp/ϕu,pred.-1st-principles-proposed Eq.(12) Cyclic loading - Rectang. sections 89 1.28 0.995 69.5% θu,exp/θu,Eq.(5) Cyclic. Conforming members w/ rect. web. Eq.(10),(12),(13) 888 1.125 1.005 54.9% θu,exp/θu,Eq.(5) Cyclic. Conforming beams, rect. columns. Eqs.(10),(12),(14) 823 1.10 1.00 53.7% θu,exp/θu,Eq.(5) Cyclic. Conforming circular columns. Eqs.(10),(12),(15) 76 1.035 1.00 30.5% θu,exp/θu,Eq.(5) Cyclic. Nonconform. rect. columns, walls Eqs.(10),(12),(16) 36 0.995 0.995 39.4% θu,exp/θu,Eq.(5) Monotonic. Beams, rectangular columns. Eqs.(10),(11),(17) 276 1.35 1.005 93% θu,exp/θu,Eq.(5) Monotonic. Beams, rectangular columns. Eqs.(10),(11),(18) 276 1.36 1.00 94.1% θu,exp/θu,Eq.(5) Cyclic. Conforming beams, rect. columns, walls. Eq.(8),(19) 888 1.10 0.995 56.4% θu,exp/θu,Eq.(5) Cyclic. Conforming beams, rectangular columns. Eq.(8),(20) 823 1.09 1.00 52.5% θu,exp/θu,Eq.(5) Cyclic. Conforming circular columns. Eqs.(8),(21) 76 1.10 0.995 33.8% θu,exp/θu,Eq.(5) Cyclic. Non-conforming rectang. columns, walls. Eq.(8),(22) 36 1.12 1.10 41.4% θu,exp/θu,Eq.(23) Cyclic. Conforming beams, rectan. columns, walls. Eq.(23) 880 1.025 0.995 39.0% θu,exp/θu,Eq.(23) Monotonic. Conform. beams, rect. columns, walls. Eq.(23) 279 1.095 1.005 55.4% θu,exp/θu,Eq.(23) Monotonic or cyclic. Conforming walls. Eq.(23) 59 0.98 0.995 30.9% θu,exp/θu,Eq.(23) Monotonic or cyclic. Conf. beams, columns, walls Eq.(23) 1159 1.045 1.00 43.9% θu,exp/θu,Eq.(23) Cyclic. Non-conforming rectang. columns, walls. Eq.(23) 36 0.855 0.835 34.8% θu,exp/θu,Eq.(24) Cyclic. Conforming beams, rectan. columns, walls. Eq.(24) 880 1.05 1.00 39.6% θu,exp/θu,Eq.(24) Monotonic. Conform. beams, rect. columns, walls. Eq.(24) 279 1.135 1.025 54.4% θu,exp/θu,Eq.(24) Monotonic or cyclic. Conforming walls. Eq.(24) 59 1.02 1.005 28.9% θu,exp/θu,Eq.(24) Monotonic or cyclic. Conf. beams, columns, walls. Eq.(24) 1159 1.075 1.00 44.7% θu,exp/θu,Eq.(24) Cyclic. Non-conforming rectang. columns, walls. Eq.(24) 36 0.875 0.865 34.2% VR,exp/VR,Eq.(25) Ductile shear. Columns with rectangular section. Eq.(25) 146 0.995 0.99 15% VR,exp/VR,Eq.(25) Ductile shear. Columns with circular section. Eq.(25) 45 1.045 1.005 17.0% VR,exp/VR,Eq.(25) Ductile shear. Walls with rectangular or T-section. Eq.(25) 6 0.99 0.99 5.7% VR,exp/VR,Eq.(25) Ductile shear. Piers with hollow rectang. section. Eq.(25) 11 1.10 0.995 16.2% VR,exp/VR,Eq.(25) Ductile shear. Columns, walls or piers. Eq.(25) 208 1.01 0.995 15.6% VR,exp/VR,Eq.(26) Ductile shear. Columns with rectangular section. Eq.(26) 146 1.00 1.00 13.5% VR,exp/VR,Eq.(26) Ductile shear. Columns with circular section. Eq.(26) 45 1.015 0.975 16.2% VR,exp/VR,Eq.(26) Ductile shear. Walls with rectangular or T-section, Eq.(26) 6 1.08 1.06 12% VR,exp/VR,Eq.(26) Ductile shear. Piers with hollow rectang. section, Eq.(26) 11 1.155 1.05 15.3% VR,exp/VR,Eq.(26) Ductile shear. Columns, walls or piers, Eq.(26) 208 1.015 1.00 14.6% VR,exp/VR,Eq.(29) Web crushing. Walls with rectangular or T-section, Eq.(29) 37 1.02 1.00 12.9%

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4. ULTIMATE DRIFT (OR CHORD ROTATION) FOR FLEXURE-CONTROLLED FAILURE 4.1 Introduction Specimens that failed after yielding in flexure were distinguished from those failing in shear before flexural yielding, on the basis of the following two criteria: (a) the reported behaviour before and at failure and (b) a comparison of the measured lateral resistance with the resistance derived from the value of the yield moment, My, calculated from first principles (taking into account the scatter exhibited in Figure 3). Specimens characterised in this way as yielding in flexure, were further categorised as ultimately failing in flexure or in shear on the basis of the reported experimental behaviour at failure. When the available information was incomplete, unclear or unconvincing, recourse was also made to comparisons of the measured ultimate deformation with the predictions of models such as those described in the rest of this paper for flexure- or shear-controlled ultimate behaviour. 4.2 Formulations based on curvatures and plastic-hinge length If inelastic behaviour and failure is controlled by flexure, the familiar description of the plastic component of drift ratio (or chord rotation) over the shear span Ls as the product of the plastic component of (the ultimate) curvature, ϕu-ϕy, and a plastic-hinge length, Lpl, is very appealing:

−−+=+=

s

plplyuy

pluyu L

LL

5.01)( φφθθθθ (5)

Notwithstanding its mechanical and physical appeal, the real criterion for the value of Eq.(5) is its ability to predict the experimental ultimate drift ratio or chord rotation, θu. Empirical expressions for Lpl needed to this end cannot be developed independently of the models used for the other variables entering into Eq.(5), namely for θy, ϕu andϕy. To maintain the apparent rationality of Eq.(5), priority should be given to models based on rational mechanics. For ϕy, the model based on first principles and outlined in Section 3 is a natural choice, despite the unsatisfactory agreement with “measured” values displayed in Figure 2. The natural choice for ϕu is a model with similar basis as that for ϕy, namely the plane sections hypothesis and equilibrium, but with nonlinear σ-ε laws. [Panagiotakos and Fardis, 2001] presented analytical expressions for ϕu for sections with rectangular compression zone, unsymmetric reinforcement concentrated at the two flanges and uniformly distributed (web) reinforcement in-between. These expressions are based on: (a) a σ-ε law for steel taken as elastic-perfectly plastic for relatively low steel strains, such as those associated with section ultimate conditions due to crushing of the concrete, or elastic-linearly strain-hardening from fy at εy to the ultimate strength ft at strain εsu, for the large steel strains typical of section failure due to steel rupture; (b) a σ-ε law for concrete which is parabolic up to ultimate strength and horizontal thereafter up to the ultimate concrete strain. The expressions for ϕu take into account spalling of the unconfined concrete cover and confinement of the concrete inside the hoops thereafter. The only important open parameter is the model to be used for the ultimate strength, fcc, the associated strain, εco,c and the ultimate strain of concrete, εcu,c, as these are affected by confinement. The value of ϕu is sensitive mainly - if not only - to the ultimate strain of confined concrete, εcu,c. The options considered for the confinement model are: a) The original Mander model [Mander et al, 1988]. It comprises: (a) an increase of fc and εco with

confining pressure p, which is in good agreement with triaxial test results on confined concrete in concentric compression and (b) a concrete ultimate strain derived from a postulated conservation of strain energy, giving, if the σ-ε law of concrete (confined or not) is taken as horizontal after ultimate strength:

( )ccywswsucuccu ff /2,, ρεεε += (6)

where εcu may be taken equal to 0.004 and εsu,w is the ultimate strain of transverse reinforcement. b) The Mander model, with Eq.(6) as modified by Priestley [Paulay and Priestley, 1992]:

518


( )ccywswsuccu ff /24.1004.0 ,, ρεε += (7)

c) The model in CEB/FIP Model Code 90 [Comité Eurointernational du Beton, 1993]. This is the reference model in Europe for confinement, as it has been adopted in the 2003 version of Eurocode 2 (the European concrete design standard) and, therefore, by Eurocode 8 (the European seismic design standard) as well. This model provides for a more modest increase of fc and εco with p, and for an ultimate strain εcu,c of:

cywsccuccu fffp /2.00035.0/2.0, αρεε +=+= (8)

where ρs is the transverse reinforcement ratio (minimum among the two transverse directions), fyw its yield stress and α the confinement effectiveness factor according to [Sheikh and Uzumeri, 1982]:

−

−

−= ∑

cc

i

c

h

c

hhbb

hs

bs

61

21

21

2α (9)

with sh the centerline spacing of stirrups, bc and hc the dimensions of the confined core to the inside of the hoop and bi the centerline spacing along the perimeter of the cross-section of those longitudinal bars (indexed by i) which are laterally restrained by a stirrup corner or by a cross-tie.

d) The following expression for the strength of confined concrete, giving slightly lower strength enhancement than the Mander model:

+=

87.0

7.31c

ywsccc f

fff

αρ (10)

along with the following variations of Eqs.(6) or (7):

( )ccywswsuccu ff /26.0004.0 ,, ρεε += (11) or

( )ccywswsuccu ff /25.1004.0 ,, αρεε += (12) As shown in Figure 4 and 5 and by the statistics of the ratio of experimental to predicted values in Table 4, in members with rectangular section confinement model option (d) and Eqs.(11) and (12) provide a better average fit to the measured values of ϕu in monotonic or cyclic tests, respectively, than the other three alternatives, and with less scatter. Alternative (a), with Eq.(6) derived from the original Mander model, provides almost the same average agreement as confinement model option (d), especially with monotonic data, albeit with significantly larger scatter. Considering the comparison with test results as a vindication of confinement model (d) and of Eqs.(11) and (12) (whatever the value of the measured data may be, in view of the scatter and the uncertainty introduced by the gauge length), empirical expressions are derived for the plastic hinge length, Lpl, to fit Eq.(5) to the data on ultimate chord rotation, θu, of all members failing in flexure, using the value of ϕu derived from first principles and confinement model option (d) above. The value of ϕy used in Eq.(5) is also the one derived from first principles. It was found that a better overall fit of Eq.(5) to the data on θu, is possible, if Eqs.(2)-(4) are used for the chord at yielding, θy, instead of their 1st (flexural) term alone. It was also found that the same expression for Lpl cannot fit both the monotonic and the cyclic data. Moreover, for cyclic loading different expressions are appropriate for members with or without detailing for earthquake resistance (often called “conforming” vs. “non-conforming” detailing, also in Tables 1 and 2). Expressions tried for Lpl are linear combinations of the shear span, Ls, and/or of the section depth, h (or D in circular sections). A term proportional to the product of the diameter of the tension reinforcement, db, and its yield stress, fy, is added, to account for the effect of slippage of the longitudinal reinforcement from its anchorage zone beyond the section of maximum moment. This term is multiplied with asl, where asl = 0 if such slippage is not physically possible, while asl = 1 if it is. A term inversely proportional to

cf , as in the 3rd term of Eqs.(2)-(4) for dependence on ultimate bond stress, is not to advantage.

519


Mander

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

φu,pred (1/m)

φu,

exp

(1/m

)

median: φu,exp=1.04φu,pred

(a)

Priestley

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

φu,pred (1/m)

φu,

exp

(1/m

)


(b)

CEB/FIP, EC2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

φu,pred (1/m)

φu,

exp

(1/m

)


(c)

Proposed model0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

φu,pred (1/m)

φu,

exp

(1/m

)median: φu,exp=φu,pred

(d)Figure 4 Measured ultimate curvatures in monotonic or cyclic tests of members with rectangular section, compared to predictions derived from 1st principles using various confinement models: (a)

Mander model, Eq.(6); (b) Priestley model, Eq.(7); (c) CEB/FIP MC90, Eqs.(8),(9); (d) Eqs. (10)-(12) The expressions found to provide the best overall fit to θu for members failing in flexure are: For “conforming” beams, columns and walls with rectangular web, under cyclic loading (see Figure 5(a)):

by

slsLhcypl dMPaf

ahLL50

)(13.0026.0,, ++=

(13) Alternative expression for “conforming” beams and columns with rectangular section (not for walls), under cyclic loading (see Figure 5(b)):

by

slhcypl dMPaf

ahL60

)(3.0,, +=

(14) For “conforming” columns with circular section, under cyclic loading (see Figure 6(a)):

by

slscycirpl dMPaf

aLL80

)(175.0,, +=

(15)

520


0

5

10

15

20

0 5 10 15 20Θu,pred (%)

Θu,

exp(

%)

beams&columnswalls median: θu,exp=θu,pred

5% fractile: θu,exp=0.37θu,p

(a)

0

5

10

15

20

0 5 10 15 20Θu,pred (%)

Θu,

exp(

%)

median: θu,exp=θu,pred

5% fractile: θu,exp=0.37θu,pred

(b)Figure 5 Comparison of experimental ultimate chord rotations in cyclic tests to the predictions of

Eq.(5) for confinement according to Eqs.(10),(12): (a) “conforming” beams, columns or walls with rectangular section, plastic hinge length per Eq.(13);

(b) “conforming” beams and columns with rectangular section, plastic hinge length from Eq.(14).

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 1

Θu,pred (%)

Θu,

exp

(%)

6


5% fractile: θu,exp=0.525θu,median

(a)

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 1

Θu,pred (%)

Θu,

exp

(%)

6


5% fractile: θu,exp=0.6θu,median

(b)Figure 6 Comparison of experimental ultimate chord rotation in cyclic tests of columns with circular section, with the predictions of Eq.(5):(a) for confinement according to Eqs.(10),(12) and plastic hinge length from Eq.(15); or (b) for confinement according to Eq.(8) and plastic hinge length from Eq.(21) For “non-conforming” columns and walls with rectangular section, under cyclic loading (see Figure 7(a)):

by

slsoldpl dMPaf

ahLL100

)(125.0025.0, ++= (16)

For beams and columns with rectangular section (not walls), “conforming” or not, under monotonic loading (see Figure 8(a)):

by

slsLhmopl dMPaf

ahLL40

)(5.007.0,, ++= (17)

or, as an almost equivalent alternative (see Figure 8(b)):

by

slhmopl dMPaf

ahL32

)(8.0,, += (18)

521


0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16Θu,pred (%)

Θu,

exp (

%)

median θu,exp=θu,pred

5% fractile θu,exp=0.4θu,media

(a)

CEB

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16Θu,pred (%)

Θu,

exp (

%)

median θu,exp=1.1θu,pred

5% fractile θu,exp=0.4θu,median

(b)Figure 7 Comparison of experimental ultimate chord rotation in cyclic tests of columns or walls with

rectangular section and “non-conforming” detailing, with the predictions of Eq.(5): (a) for confinement according to Eqs.(10),(12) and plastic hinge length from Eq.(16);

(b) for confinement according to Eq.(8) and plastic hinge length from Eq.(22)

0

5

10

15

20

25

30

0 5 10 15 20 25 30Θu,pred (%)

Θu,

exp(

%) median: θu,exp=θu,pred


(a)

0

5

10

15

20

25

30

0 5 10 15 20 25 30Θu,pred (%)

Θu,

exp(

%) median: θu,exp=θu,pred


(b)Figure 8 Comparison of experimental ultimate chord rotations in monotonic tests of beams or

columns with rectangular section, “conforming” or not, to the predictions of Eq.(5) for confinement according to Eqs.(10),(11) and plastic hinge length from: (a) Eq.(17); or (b) Eq.(18).

Because option (c) (i.e. the model in CEB/FIP Model Code 90) is now the reference model in Europe for confinement - as it has been adopted in Eurocode 2 and, therefore, is the basis for Eurocode 8 as well - expressions parallel to the ones above are developed for Lpl, for use in Eq.(5) along with the value of ϕu derived from first principles and confinement option (c), including Eq.(8) for the ultimate strain. For “conforming” beams, columns and walls with rectangular web, under cyclic loading (see Figure 9(a)):

by

slsLhcyCEBpl dMPaf

ahLL25

)(035.006.0,,, ++= (19)

Alternative for “conforming” beams and columns with rectangular section (not for walls), under cyclic loading (see Figure 9(b)):

by

slhcyCEBpl dMPaf

ahL38

)(5.0,,, += (20)

522


0

5

10

15

20

25

30

0 5 10 15 20 25 30Θu,pred (%)

Θu,

exp(

%)

beams&columns

walls median: θu,exp=θu,pred


(a)

0

5

10

15

20

25

30

0 5 10 15 20 25 30Θu,pred (%)

Θu,

exp(

%)



(b)Figure 9 Comparison of experimental ultimate chord rotations in cyclic tests to the predictions of

Eq.(5) for confinement according to Eq.(8) (CEB/FIP MC90): (a) “conforming” beams, columns or walls with rectangular section, plastic hinge length per Eq.(19);

(b) “conforming” beams and columns with rectangular section, plastic hinge length from Eq.(20). For “conforming” columns with circular section, under cyclic loading (see Figure 6(b)):

by

slscycirCEBpl dMPaf

aDLL42

)(35.0135.0,,, ++= (21)

It proved to be almost impossible to find an expression for Lpl providing good average fit to the results of cyclic tests on “non-conforming” columns and walls with rectangular section, when confinement option (c) is used for the calculation of ϕu. The best expression found, Eq.(22) below, underestimates test results on average by 10% (see Figure 7(b)):

by

slsoldCEBpl dMPaf

aLL10

)(2.0,, += (22)

Simirarly, no expression could be found for Lpl that provides an acceptable fit to the monotonic test results, when calculation of ϕu is based on the CEB/FIP confinement model (option (c), Eq.(8)). The search exercise for expressions for Lpl that provide satisfactory average fit of Eq.(5) to the chord rotation at flexural failure, θu, has shown that such fitting is associated with large scatter. The scatter is reflected by the large coefficients of variation listed in Table 4. It is also evident from Figures 5 to 9, which also show the line corresponding to the 5%-fractile of the experimental value, given its prediction from Eq.(5). The larger the scatter, the lower is the 5%-fractile line. Figures 5 and 9 show also a tendency for overprediction of cyclic data in “conforming” members with rectangular section, while Figure 8 shows certain underprediction of high experimental values in monotonic loading. Circular columns are a notable exception regarding scatter. For them Eq.(5), along with confinement options (d) or (c) and Eqs.(12) and (15) or (8) and (21), respectively, provide acceptable scatter. Another lesson learned from this exercise is that the expressions for Lpl in Eqs.(13)-(22) are not the only possible answer for each particular case. Other linear combinations of dbfy, Ls and/or h (or D) may provide an almost equally good average fit to the data, albeit usually with larger scatter. A third lesson is that, it is often feasible to achieve almost equivalent final fits with ϕu-values calculated from very different confinement models (obviously using each time the appropriate expression for Lpl), no matter the agreement (or lack of it) between the calculated ϕu values and the experimental values.

523


Monotonic

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Θu,pred (%)

Θu,

exp

(%)


5% fractile (all data): θu,exp=0.42θu,pred

Cyclic

0

2.5

5

7.5

10

12.5

0 2.5 5 7.5 10

Θu,pred (%)

Θu,

exp

(%)

12.5

median (all data) θu,exp=θu,pred


Walls

0

1

2

3

4

5

6

0 1 2 3 4 5Θu,pred (%)

Θu,

exp

(%)

6

median (all data): θu,exp=θu,pred


All data

0

5

10

15

20

25

30

0 5 10 15 20 25 30Θu,pred (%)

Θu,

exp(

%)

cyclic

walls

monotonic



Figure 10 Comparison of experimental ultimate chord rotation of beams, columns or walls with rectangular compression zone and “conforming” detailing, with the predictions of Eq.(23)

4.3 Empirical formulations Except for circular columns, the predictive ability of Eq.(5) - along with the corresponding expressions for Lpl - was found to be unsatisfactory. As an alternative, empirical expressions are sought for the chord rotation at flexural failure, θu, of members other than circular columns, and are developed via methods of statistics. The earlier work by [Panagiotakos and Fardis, 2001], based on about two-thirds of the present data, provides the baseline. That work has pointed out the necessity of using the monotonic and the cyclic data together, as well as the primary variables on which θu depends and a possible functional dependence. One of the main conclusions was that θu depends on whether loading to failure is monotonic or fully-reversed (cyclic), but is rather insensitive to the number of major deflection cycles preceding failure. Two alternative - and almost equivalent - expressions are developed here for the chord rotation at flexure-controlled failure, θu, of members with rectangular cross-section (or rather rectangular compression zone) and “conforming” detailing. The first, Eq.(23), is for the total ultimate chord rotation, θu, while the other, Eq.(24), is for its plastic component, θu

pl = θu-θy, with the elastic component, θy, given by Eqs.(2) or (3).

524


Monotonic

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Θu,pred (%)

Θu,

exp

(%)



Cyclic

0

2.5

5

7.5

10

12.5

0 2.5 5 7.5 10 12.5

Θu,pred (%)

Θu,

exp

(%)

median (all data) θu,exp=θu,pred


Walls

0

1

2

3

4

5

6

0 1 2 3 4 5 6Θu,pred (%)

Θu,

exp (

%)



All data

0

5

10

15

20

25

30

0 5 10 15 20 25 30Θu,pred (%)

Θu,

exp(

%)

cyclic

walls

monotonic



Figure 11 Comparison of experimental ultimate chord rotation of beams, columns or walls with rectangular compression zone and “conforming” detailing, with the predictions of Eqs.(2), (3) and (24)

( ) ( ) ( )( )

dc

yws f

f

scwallslcystu h

Lfaaa ραρ

ν

ωωαθ 100

4.0175.0

25.125,01.0max

',01.0max3.08315.01)4.01(

−+−= (23)

( ) dc

yws f

f

scwallslcy

plst

plu h

Lfaaa ραρ

ν

ωωαθ 100

375.0225.0

3.125),01.0max()',01.0max(2.0)4.01)(55.01)(45.01(

−+−= (24)

where: ast and apl

st: coefficients for the type of steel, equal to ast = 0.0194 and aplst = 0.015 for ductile hot-rolled or

for heat-treated (tempcore) steel and to ast = 0.0125 and aplst = 0.0065 for cold-worked steel;

acy: zero-one variable for type of loading, equal to 0 for monotonic loading and to 1 for cyclic loading; asl: zero-one variable for slip, equal to 1 if there is slip of the longitudinal bars from their anchorage

beyond the section of maximum moment, or to 0 if there is not (cf. Eqs.(2)-(4), (13)-(22)); awall: zero-one variable for walls, equal to 1 for shear walls and to 0 for beams or columns; ν=N/bhfc (with b = width of compression zone, N = axial force, positive for compression); ω, ω': mechanical reinforcement ratios, ρfy/fc, of longitudinal reinforcement which is in tension (including

525


web reinforcement between the two flanges) or in compression, respectively; 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 according to [Sheikh and Uzumeri, 1982], given by Eq.(9); ρd: steel ratio of diagonal reinforcement in each diagonal direction. Figures 10 and 11 show the fit of Eqs.(23) and (24) to the data for members with “conforming detailing”, and the lines corresponding to the 5%-fractile of the predictions for the full set of experimental data (1159 tests) from which Eqs.(23), (24) derive. The 5%-fractile for the plastic part of θu alone, θu

pl, is one-third of the predictions of Eq.(24). The statistics of the fitting listed in Table 4 suggest that Eqs.(23) and (24) are practically equivalent in terms of predictive ability, with a slight overall advantage of Eq.(23). Figure 12 compares the data for members with “non-conforming” detailing to the predictions of Eqs.(23) and (24), with the confinement effectiveness factor α taken equal to zero, if the stirrups are not closed with 135o hooks. These specimens are too few to support an independent statistical analysis. Their qualitative difference from the “conforming” ones prevented their inclusion in the data set that yielded Eqs.(23), (24) (whereas they were included in the analysis that yielded Eqs. (2) and (3), as they were not considered to have fundamentally different pre-yield behaviour). According to Figure 12 and to the statistics listed in Table 4 for these specimens, the predictions of Eqs.(23), (24) should be divided by 1.2 or by 1.15, respectively, to yield the expected ultimate chord rotation for “non-conforming” detailing. Taking into account the small size of the data set, the scatter about the median is considered about the same as that of the cyclic data on “conforming” members. As a result, the lines corresponding to the 5%-fractile of the predictions for the full set of experimental data for members with “conforming” detailing (1159 tests) are taken to apply also to the ones with “non-conforming” detailing.

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7Θu,pred (%)

Θu,

exp

(%)

8

median: θu,exp=θu,pred /1.2

5% fractile (all data): θu,exp=0.42θu,median

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7Θu,pred (%)

Θu,

exp

(%)

8

median: θu,exp=θu,pred /1.15

5% fractile (all data): θu,exp=0.44θu,median

Figure 12 Comparison of experimental ultimate chord rotation of columns or walls with rectangular cross section and “non-conforming” detailing, with the predictions of (a) Eq.(23) (b) Eqs.(2), (3) and

(24)

526


5. DRIFT (OR CHORD ROTATION) AT SHEAR FAILURE AFTER FLEXURAL YIELDING Concrete members that yield first in flexure, but their ultimate failure under cyclic loading clearly shows strong effects of shear, are considered to exhibit a “ductile shear” failure mode [Kowalsky and Priestley, 2000], as opposed to a “brittle shear” mode, in which failure takes place at relatively low deformations before flexural yielding. “Ductile shear” failure is controlled by diagonal tension and yielding of web reinforcement, rather than by web crushing. It has by now prevailed to quantify this failure mode via a shear resistance VR, (as this is controlled by web reinforcement according to the well-established Mörsch truss analogy) that decreases with the (displacement) ductility ratio under cyclic loading [Ascheim and Moehle, 1992, Kowalsky and Priestley, 2000, Moehle et al, 2001]. As the number of available cyclic tests that led to “ductile shear” failure is not sufficient to support development of an independent (statistical or mechanical) model for the deformation capacity of R/C members as affected or controlled by shear, the present work also adopts the solid base of the Mörsch analogy for shear, to describe in force terms a failure mode which is controlled by deformations. More specifically, the formulations proposed in [Kowalsky and Priestley, 2000] and in [Moehle et al, 2001] on the basis of a limited number of tests on circular or rectangular columns, respectively, are extended herein, using a much larger data set of “ductile shear” failures of columns with both types of section and walls. The outcome of the present effort is two models for VR, as a function of the plastic chord rotation (or displacement) ductility ratio, µpl

θ, defined as the ratio of the post-elastic chord rotation at “ductile shear” failure, to the chord rotation at yielding, θy, as this is computed from Eqs.(2)-(4). (In the application of the models for predictive purposes, the post-elastic chord rotation at “ductile shear” failure will be obtained by subtracting from the total chord rotation the value of θy from Eqs.(2)-(4). In the development of the models, though, the experimental yield chord rotation was subtracted from the total experimental value, to eliminate parasitic elastic flexibility effects observed in some tests). In both models the effect of axial force, N, on VR is accounted for through a separate term, as in [Comité Eurointernational du Beton, 1993, Kowalsky and Priestley, 2000]. A 45o truss inclination is considered, as in [Moehle et al, 2001], because truss inclinations other than 45o are normally taken when only the web reinforcement is considered to contribute to VR, (Vw term), without a separate concrete contribution (Vc term). In the first model, Eq.(25), only the Vc term decreases with µpl

θ, as in [Kowalsky and Priestley, 2000], while in the second, Eq.(26), both the Vc and the Vw terms are taken to decrease with µpl

θ, as in [Moehle et al, 2001]:

( ) ( )( ) wccs

totpl

ccs

R VAfhLfAN

LxhV +

−−⋅+−

= ,5min16.01)100,5.0max(,5min095.0116.055.0,min2

ρµθ (25)

( ) ( )( )

+

−−⋅+

−= wcc

stot

plcc

sR VAf

hLfAN

LxhV ,5min16.01)100,5.0max(,5min055.0116.055.0,min

2ρµθ

(26) where: h: depth of cross-section (equal to the diameter D for circular sections); x: compression zone depth; N: compressive axial force (positive, taken as zero for tension); Ls/h=M/Vh: shear span ratio at member end; Ac: cross-section area, taken equal to bwd for cross-sections with rectangular web of width (thickness)

bw and structural depth d, or to πDc2/4 (where Dc is the diameter of the concrete core to the inside

of the hoops) for circular sections; fc: concrete strength (ΜPa); ρtot: total longitudinal reinforcement ratio; Vw: contribution of transverse reinforcement to shear resistance, taken equal to:

a) for cross-sections with rectangular web of width (thickness) bw:

ywwww zfbV ρ= (27)

where ρw and fyw are the ratio and the yield stress of transverse reinforcement and z is the length of the internal lever arm (taken equal to d-d’ in beams and columns, or to 0.75h in walls),

527


b) for circular cross-sections:

)2(2

cDfsAV yw

h

sww −=

π (28)

where Asw is the cross-sectional area of a circular stirrup, sh is the centerline spacing of stirrups and c the concrete cover to reinforcement.

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

Vpred (kN)

Vex

p (k

N)

Rectangular Circular w alls&hollow

(a)

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

Vpred (kN)

V exp

(kN)


(b)Figure 13 Comparison of experimental shear strength in “ductile shear” failure with the predictions

of: (a) Eq.(25); (b) Eq.(26)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 8 9 10 11 12

ductility ratio (µ)

V exp

/Vpr

ed


(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 8 9 10 11 12ductility ratio (µ)

pp


(b)Figure 14 . Ratio of experimental to predicted shear strength for “ductile shear” failure, as a function

of (total) chord rotation ductility ratio at failure, for predictions from: (a) Eq.(25); (b) Eq.(26) In the fitting of Eqs.(25), (26), the experimental shear strength is taken equal to the shear force at flexural yielding, and not to the exact value of lateral force resistance at the instant “ductile shear” failure is taken to occur (In the application of Eqs.(25), (26) for the prediction of “ductile shear” failure, their right-hand-side is set equal to the - theoretical - value of My/Ls). This is also the experimental shear strength compared to the predictions of Eqs.(25), (26).in Figures 13 and 14 and near the bottom of Table 4 Eqs.(25), (26) are practically equivalent: Eq.(25) has slightly better overall statistics, while Eq.(26) gives better average agreement to the data for each one of the four types of members included in the fitting. With very few exceptions, the experimental value of θu in the 208 tests which were considered as controlled by shear and were used for the fitting of Eqs.(25), (26) is less than the flexure-controlled ultimate chord rotation predicted from Eqs.(23) or (24), and, as a matter of fact, significantly less. To pursue the possibility of using Eqs.(25), (26) as a means to predict the shear flexure-controlled ultimate chord rotation, their right-hand-side was set equal to the the theoretical value of My/Ls and Eqs.(25), (26) were solved for µpl

θ. Figure 15, which compares the resulting values of µθ = µplθ+1 to the experimental

ones, clearly shows that, despite the relatively low scatter associated with them, Eqs.(25) and (26) do not

528


lend themselves for a meaningful prediction of a shear-controlled deformation capacity. The reason for this failure is the low sensitivity of VR to µpl

θ.

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9µ∆,,exp

µ ∆,,p

red

10


(a)

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9µ∆,,exp

µ ∆,,p

red

10


(b)Figure 15 Experimental ductility ratio, compared to value predicted by inverting:

(a) Eq.(25); (b) Eq.(26) It is reminded that “ductile shear” failure, described by Eqs.(25), (26) is considered to be associated with diagonal tension and with yielding of the web reinforcement. The large majority (27) of 33 walls in the database considered to have failed in shear after having yielded in flexure, did so by web crushing at a shear force normally lower than the predictions of Eqs.(25) or (26) and at a chord rotation much less than the value corresponding to flexure-controlled failure according to Eqs.(23), (24). It is noteworthy that the shear resistance of these 27 walls, as well as that of the 10 cyclically loaded ones that experienced web crushing before yielding in flexure, seems to be insensitive to the magnitude of the deformation at which web crushing took place (see also Figure 16) and to follow the expression:

)'(1.01)100(94165.01095.0, ddbf

hL

dfbNV wc

stot

cwwebR −

−

+

+= ρ (29)

Despite the low scatter associated with Eq.(29), the data behind it are not sufficient to support proposing it as an upper limit for the shear strength of walls under cyclic loading. Nonetheless, the low magnitude of its results, compared to those given by current code rules is disconcerting. The - normalized by bwdfc - shear resistance of the few short columns in the database that fail by web crushing after flexural yielding is about double that obtained from Eq.(29), and decreases with µpl

θ Those data, though, are even fewer.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7

ductility ratio (µ)

V exp

/Vpr

ed

8

Figure 16 Ratio of experimental shear strength to prediction of Eq.(29) in shear walls (rectangular, T or barbelled) failing by web crushing, as a function of (total) chord rotation ductility ratio at failure

529


530

6. ACKNOWLEDEMENTS The European Commission provides financial support to this work under project SPEAR (Seismic Performance Assessment and Rehabilitation, contract No: G6RD-CT2001-00525) of its GROWTH programme. Prof. Otani participates in this project, within its component for international collaboration. The contributions of Dr. T.B. Panagiotakos to the on-going development of the database, as well to earlier phases of this research, are gratefully acknowledged. 7. REFERENCES ASCE (2000), Prestandard for the seismic rehabilitation of buildings. American Society of Civil Engineers (FEMA Report 356), Reston, Va. ASCE, (2001), Seismic evaluation of existing buildings. ASCE draft Standard, American Society of Civil Engineers, Reston, Va. Ascheim, M.A. and J.P.Moehle(1992), "Shear Strength and Deformability of RC Bridge Columns Subjected to Inelastic Cyclic Displacements" University of California, Earthquake Engineering Research Center, Report UCB/EERC-92/04, Berkeley, CA Biskinis, D., G. Roupakias, and M.N. Fardis(2002), "Stiffness and Cyclic Deformation Capacity of Circular Concrete Columns", in: Befestigungstechnik Bewehrungstechnik und …Festschrift zu Ehren von Prof. Dr.-Ing. Rolf Eligehausen anlässlich seines 60. Geburtstages (W. Fuchs, H.-W. Reinhardt, eds.), Aktuelle Beitrage aus Forschung und Praxis, Ibidem-Verlag, Stuttgart, pp. 321-330. Comité Eurointernational du Beton(1993), CEB/FIP Model Code 1990. T. Telford (ed.), London. Comité Européen de Normalisation(2003a), Draft European Standard prEN1998-3:200x Eurocode 8: Design of structures for earthquake resistance. Part 3: Strengthening and repair of buildings. Revised final PT Draft (Stage 34) Doc. CEN/TC250/SC8/N371, July 2003, Brusells. Comité Européen de Normalisation(2003b), Draft European Standard prEN1998-1:200x Eurocode 8: Design of structures for earthquake resistance. Part 1: General rules, seismic actions and rules for buildings. Final Version (Stage 49), Doc. CEN/TC250/SC8/N335A, January 2003, Brusells. Federation Internationale du Beton(2003a), Seismic Assessment and Retrofit of RC Buildings fib Bulletin No.24, Lausanne. Federation Internationale du Beton(2003b), Displacement-based Design of RC Buildings Chapter 5: “Displacement Capacity of Members and Systems”, fib Bulletin No.25, Lausanne Kowalsky, M. and M.J.N. Priestley(2000), "Improved Analytical Model for Shear Strength of Circular Reinforced Concrete Columns in Seismic Regions" Structural Journal, American Concrete Institute, Vol. 97, No. 3, pp.388-396. Mander, J.B., M.J.N. Priestley, R. Park(1988), "Theoretical Stress-strain Model for Confined Concrete", Journal of Structural Engineering, American Society of Civil Engineers, V.114, No.8, pp.1827-1849. Moehle J., A. Lynn, K. Elwood, H. Sezen(2001), "Gravity Load Collapse of Building Frames during Earthquakes" 2nd US-Japan Workshop on Performance-based Design Methodology for Reinforced Concrete Building Structures. Pacific Earthquake Engineering Research Center, Richmond, CA. Panagiotakos T. and M.N. Fardis (2001), "Deformation of R.C. Members at yielding and ultimate", Structural Journal, American Concrete Institute, Vol. 98, No. 2, pp. 135-148. Paulay, T. and M.J.N Priestley(1992), Seismic Design of Reinforced Concrete and Masonry Buildings, J. Wiley, New York, N.Y. Sheikh, S.A. and S.M. Uzumeri(1982), "Analytical Model for Concrete Confinement in Tied Columns", Journal of Structural Division, American Society of Civil Engineers V.108, No. ST12, pp.2703-2722.

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