deformations of concrete members at ......over 300 uniaxial tests are identified to fail in shear...

U N I V E R S I T Y O F P A T R A S DEPARTMENT OF CIVIL ENGINEERING

Report Series in Structural and Earthquake Engineering

DEFORMATIONS OF CONCRETE MEMBERS AT YIELDING AND ULTIMATE UNDER MONOTONIC OR CYCLIC LOADING

(INCLUDING REPAIRED AND RETROFITTED MEMBERS)

DIONYSIS BISKINIS, MICHAEL N. FARDIS

Report No. SEE 2009-01

January 2009

i

DEFORMATIONS OF CONCRETE MEMBERS

AT YIELDING AND ULTIMATE UNDER MONOTONIC OR CYCLIC LOADING

(INCLUDING REPAIRED AND RETROFITTED MEMBERS)

by

DIONYSIS BISKINIS

Post-Doctoral Researcher

And

MICHAEL N. FARDIS

Professor

The report is printed with the

financial support of the European Commission

under FP7 project A.C.E.S.

Any opinions, findings, and conclusions or recommendations

expressed in this material are those of the author(s)

and do not necessarily reflect those of the European Commission.

Report No. SEE 2009-01

Department of Civil Engineering, University of Patras

January 2009

iii

Abstract

Models are developed/calibrated for the moment, the chord rotation and the

secant stiffness at flexural yielding, as well as the ultimate deformation of beams,

rectangular columns or walls and members of T-, H-, U- or hollow rectangular section, on

the basis of a databank of tests on RC members with continuous bars. Simple criteria are

developed for the identification of adverse shear effects on the yield moment or the

ultimate deformation. The models for the moment and the chord rotation at flexural

yielding apply only to members whose yield moment is not reduced by shear effects. The

models developed employ simple, explicit expressions suitable for practical application,

without moment-curvature analysis. They are extended to members with bars lap-spliced

starting at the end section, as well as to those retrofitted with Fibre-Reinforced-Polymer

(FRP) or concrete jackets before or after pre-damage by cyclic loading. The effects of

biaxial loading on member yielding and ultimate deformation are examined.

The proposed models have been adopted in Part 3 of Eurocode 8 for the seismic

assessment and retrofitting of existing concrete buildings, or represent an advancement

over models proposed earlier by the authors and therein.

v

Table of Contents

Abstract ........................................................................................................................................................................... iii

1 Introduction ....................................................................................................................................................... 1

2 Deformations at flexural yielding of members with continuous bars ............................................................ 5

2.1 Uniaxial yield moment and curvature of section with rectangular, T- or L- compression zone ............. 5

2.2 Fixed-end rotation due to pull-out of longitudinal bars from their anchorage beyond the end section

at uniaxial yielding ................................................................................................................................................... 10

2.3 Chord rotation and secant stiffness to yield point in uniaxial flexure for members of rectangular, T-,

H-, U-, or hollow rectangular section ..................................................................................................................... 11

2.4 Yield moment, chord rotation and secant stiffness to yield point under biaxial flexure ....................... 15

3 Effect of lap-splicing of longitudinal bars in the plastic hinge zone on member flexure-controlled

yielding and secant stiffness to yield point ............................................................................................................. 17

3.1 Yield moment and curvature of section with rectangular, T- or L- compression zone ......................... 17

3.2 Chord rotation and effective stiffness at yielding ...................................................................................... 20

4 Flexure-controlled ultimate deformations of members with continuous bars ............................................ 23

4.1 Uniaxial ultimate curvature of members with rectangular compression zone ....................................... 23

4.2 Strains of steel and concrete at section ultimate curvature ...................................................................... 27

4.3 Fixed-end rotation due to pull-out of longitudinal bars from their anchorage beyond the end section

at member flexural failure ...................................................................................................................................... 30

4.4 Uniaxial ultimate chord rotation from curvatures and the plastic hinge length .................................... 31

4.5 Empirical uniaxial ultimate chord rotation ............................................................................................... 32

4.6 Flexure-controlled ultimate chord rotations in biaxial loading ............................................................... 37

5 Effect of lap-splicing of longitudinal bars in the plastic hinge zone on flexure-controlled ultimate

deformations ............................................................................................................................................................. 38

6 Shear Strength after Flexural Yielding .......................................................................................................... 42

6.1 Introduction.................................................................................................................................................. 42

6.2 Models of shear resistance in diagonal tension under inelastic cyclic deformations after flexural

yielding ...................................................................................................................................................................... 43

7 RC-jacketing of columns ................................................................................................................................. 46

7.1 Introduction.................................................................................................................................................. 46

7.2 Simple rules for the strength, the stiffness and the deformation capacity of jacketed members .......... 46

8 FRP-jacketing of columns ............................................................................................................................... 52

8.1 Seismic retrofitting with FRPs .................................................................................................................... 52

8.2 FRP-wrapped columns with continuous vertical bars .............................................................................. 52

8.2.1 Yield moment and effective stiffness to yield point .................................................................................. 52

vi

8.2.2 Flexure-controlled deformation capacity ................................................................................................... 55

8.3 FRP-wrapped columns with ribbed (deformed) vertical bars lap-spliced in the plastic hinge region . 58

8.4 Cyclic shear resistance of FRP-wrapped columns .................................................................................... 61

9 Repaired concrete members............................................................................................................................ 64

REFERENCES .............................................................................................................................................................. 67

NOTATION................................................................................................................................................................... 71

1

1 Introduction

In performance-based seismic assessment and retrofitting of existing buildings [2], [11], as well as

in displacement-based seismic design, flexure-controlled components are checked by comparing

seismic deformation demands to corresponding limit deformations. This requires knowledge of the

deformations of flexure-controlled members at key points of their force-deformation response,

namely at yielding and ultimate conditions, in terms of the geometry and material properties of the

member and its reinforcement and axial load. Moreover, member seismic deformation demands to

be compared with deformation limits should be obtained using realistic effective stiffness values

for all members of the lateral load resisting system. The default stiffness values of current codes

for seismic design of new buildings overestimate member stiffness, to be on the safe-side for

force-based design; but they underestimate seismic deformation demands and are unsafe for

displacement-based evaluation or design.

In order to compare seismic deformation demands to corresponding limit deformations for

flexure-controlled members, member ultimate deformations should be known as a function of the

geometry and material properties of the member and its reinforcement, the axial load and the

detailing (lap-spliced bars, stirrups without 135o hooks, etc.). To define deformation limits for

target reliability levels, not only the mean value of ultimate deformation should be known, but also

its dispersion. Note that, members subjected to cyclic loading after yielding in flexure may

ultimately fail in shear, owing to shear strength degradation. This affects the member ultimate

deformation. So, the failure mode of a concrete member in cyclic loading, namely whether it is by

bending or in shear, and how this affects the value of the member ultimate deformation, is also a

very important parameter to be investigated.

It is still common in many parts of the world – including Europe’s seismic areas – to lap-

splice all vertical bars of columns or walls at floor levels for convenience of bar fixing. Besides,

short lap splices in vertical members at floor level are a typical deficiency in existing substandard

construction all over the world.

2

The present report contributes to the current worldwide effort to improve our capability for

estimation of key deformation properties and the effective stiffness of RC members. Its

contributions are in the form of simple, practical models applying (sometimes with minor

variations) to those beams, rectangular columns or walls and members with T-, H-, U- or hollow

rectangular section and continuous or lap-spliced bars whose yield moment is not reduced by

shear effects. They are developed/calibrated using a large database of tests [8] of concrete

members. About 2500 uniaxial tests in the database are identified as free of adverse effects of

shear on yielding and utilised further in this paper. The breakdown of these tests and the range and

mean value of the main parameters of their specimens are summarised in Table 1.1. About 1620

tests of the database are identified to have flexural behaviour until failure. Table 1.2 gives the

breakdown of these tests and the range and mean value of the main parameters of these specimens.

Over 300 uniaxial tests are identified to fail in shear after flexural yielding and are utilized to

develop/calibrate models of shear strength degradation under cyclic loading.

Table 1.1: Range and mean of parameter values in the groups of specimens in the database, with

continuous or lapped bars, yielding in flexure

Parameter

2426 specimens with continuous bars 114 members with lap-spliced ribbed

bars

rectangular beams or columns: 2084

rectangular walls: 175

non-rectangular sections: 167

beams/columns: 103


min/max mean min/max mean min/max mean min/max mean min/max mean

effective depth, d (m) 0.072 / 2.3 0.28 0.535 / 2.9 1.2 0.18 / 3.1 1.28 0.072 / 0.72 0.34 0.415 / 1.125 0.67

shear-span-to-depth ratio, Ls/h 1 / 13 4 0.6 / 5.55 1.95 0.6 / 8.33 2.46 2.75 / 8.4 4.72 1.6 / 3 2.75

section aspect ratio, h/bw 0.2 / 4 1.3 4 / 30 11.6 - - 0.2 / 2 1.2 4 / 12 6.25

fc (MPa) 9.6 / 118 37.2 18.7 / 86 36 20 / 102 41.8 16.7 / 40.8 27.5 22.5 / 38 28.2

Axial-load-ratio, N/Acfc -0.05 / 0.9 0.125 0 / 0.35 0.057 0 / 0.5 0.07 0 / 0.4 0.1 0 / 0.19 0.07

transverse steel ratio, ρw (%) 0.02 / 3.54 0.62 0 / 2.18 0.64 0.1 / 2.09 0.59 0.045 / 0.92 0.36 0.175 / 0.56 0.265

total longitudinal steel ratio ρtot (%)0.11 / 8.55 1.97 0.07 / 4.27 1.55 0.34 / 6.19 1.29 0.295 / 5.13 1.79 0.78 / 1.1 0.95

lapping-to-bar-diameter ratio lo/dbL - - - - - - 15 / 50 34 20 / 60 30

3

Table 1.2: Range and mean of parameter values in the groups of specimens in the database, with continuous or lapped bars, reaching ultimate deformation in flexure

Parameter

1619 specimens with continuous bars 92 members with lap-spliced ribbed bars 1539 specimens detailed for earthquake resistance 80 rectangular

columns with poor detailing

1395 rectangular beams/columns:

88 rectangular walls:


81 beams/columns:

11 non-rectangular sections

min/max mean min/max mean min/max mean min/max mean min/max mean min/max mean

effective depth, d (m) 0.085 / 2.3 0.29 0.62 / 2.625 1.265 0.18 / 3.1 1.26 0.072 / 1.680.363 0.072 / 0.72 0.33 0.415 / 1.125 0.67

shear-span-to-depth ratio, Ls/h 1 / 13 3.9 0.7 / 5.55 2.15 0.65 / 8.33 2.7 1.8 / 6.4 4.03 2.75 / 8.4 4.72 1.6 / 3 2.75

section aspect ratio, h/bw 0.2 / 4 1.3 4 / 28.3 9.85 - - 0.5 / 28.3 2.26 0.2 / 2 1.2 4 / 12 6.25

fc (MPa) 9.6 / 118 37.2 22 / 57 34.8 20 / 102 41.7 10.6 / 67 34.6 16.7 / 40.8 28.1 22.5 / 38 28.2

axial-load-ratio, N/Acfc -0.05 / 0.9 0.14 0 / 0.35 0.058 0 / 0.3 0.065 0 / 0.47 0.2 0 / 0.4 0.11 0 / 0.19 0.07

transverse steel ratio, ρw (%) 0.02 / 3.35 0.725 0 / 2.18 0.66 0.1 / 2.09 0.6 0.055 / 0.790.235 0.045 / 0.92 0.27 0.175 / 0.56 0.265

total longitudinal steel ratio ρtot %

0.21 / 6.52 2.04 0.15 / 4.27 1.28 0.35 / 6.19 1.3 0.07 / 3.53 1.54 0.295 / 5.13 1.92 0.78 / 1.1 0.95

lapping-to-bar-diameter, lo/dbL - - - - - - - - 15 / 50 32 20 / 60 30

The most widely used retrofitting technique for concrete columns is jacketing of their full

storey height with reinforced concrete. Wrapping of their ends with fibre-reinforced polymers

(FRPs), is coming up as an overall cost-effective technique. Design of the retrofitting requires

knowledge of important properties of the retrofitted members, such as their cyclic lateral strength,

secant stiffness to yield point and cyclic deformation capacity, as a function of the parameters of

the retrofitting. The database of test results [8] includes also 57 concrete members with rectangular

section, retrofitted with reinforced concrete jacket and about 240 members retrofitted with FRP

jacket. Table 1.3 gives the breakdown of the FRP jacketed tests and the range and mean value of

the main parameters of these specimens. Utilization of test results for the retrofitted members,

yields in modifications of the proposed models for non-retrofitted members, so that behaviour of

retrofitted members can be quantified in a similar way. The application of FRP jacket to enhance

the flexural behaviour of a concrete member with deficient lap-splice length is also examined.

An additional dataset of 33 concrete members on the database includes test results of repaired

specimens. They are utilized to examine whether the original yield moment, effective stiffness to

yield-point and deformation capacity of a concrete member are re-instated by repair.

4

Table 1.3: Range and mean of parameter values in the groups of specimens in the database with FRP jacket, with continuous or lapped bars.

Parameter 188 specimens with rectangular section

and continuous bars 44 members with rectangular section

and lap-spliced ribbed bars

min / max mean min / max mean

effective depth, d (m) 0.17 / 0.72 0.30 0.18 / 0.72 0.41

shear-span-to-depth ratio, Ls/h 0.65 / 7.41 3.59 2 / 6.59 4.21

section aspect ratio, h/bw 0.33 / 2.6 1.11 1 / 2 1.52

fc (MPa) 10.6 / 90 32.02 13.7 / 55 31.53

Axial-load-ratio, N/Acfc 0 / 0.85 0.253 0 / 0.33 0.148

transverse steel ratio, ρw (%) 0.02 / 0.715 0.215 0.082 / 0.446 0.224

total longitudinal steel ratio ρtot (%) 0.814 / 7.6 2.103 0.814 / 3.9 1.89

geometric ratio of the FRP, ρf (%) 0.01 / 5.31 0.615 0.13 / 5.31 0.752

lapping-to-bar-diameter ratio lo/dbL - - 15 / 45 29.1

5

2 Deformations at flexural yielding of members with continuous bars

2.1 Uniaxial yield moment and curvature of section with rectangular, T- or L- compression

zone

To avoid a full moment-curvature analysis, expressions for the uniaxial yield moment, My, and

curvature, φy, from plane section analysis and linear material σ-ε laws have been given in [19] for

sections with rectangular compression zone. They are extended here to T-, U- or L- compression

zones. The width and thickness of the compression flange are denoted as b and t, respectively and

the total width of the webs as bw; d is the section’s effective depth; the tension and compression

reinforcement areas, As1, As2, are normalized by bd into ratios, ρ1, ρ2, respectively. Reinforcement

(almost) uniformly distributed between the tension and the compression bars, termed “web”

reinforcement, has area Asv and is normalized by bd into a ratio of ρv. The curvature at yielding of

the tension bars is:

( )dEf

ys

yy ξ

ϕ−

=1

1 (2.1)

where fy1 is the yield stress of the tension reinforcement and ξy the neutral axis depth at yielding

(normalized to d):

( ) ABAy αααξ −+=2/122 2 (2.2)

with α=Es/Ec denoting the ratio of elastic moduli and A, B being given by:

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+

+++=⎟⎟

⎠

⎞⎜⎜⎝

⎛−+⎟

⎟⎠

⎞⎜⎜⎝

⎛+++= 1

21

2'1

',11 2

121

121

wy

v

wwyv

w bb

dt

bdfN

bbB

bb

dt

bdfN

bbA

αδρ

δρρα

ρρρ (2.3)

where δ’=d’/d, with d’ denoting the distance of the center of the compression reinforcement from

the extreme compression fibres and the axial force, N, is taken positive for compression.

A section with high axial-load-ratio, N/Acfc, may exhibit apparent yielding owing to significant

nonlinearity of the concrete in compression before the tension steel yields. It can be modeled

simply by identifying it with exceedance of a certain strain at the extreme compression fibres, with

both steel and concrete considered elastic till then. In [19] tests of members with high axial load

6

led to the following value for this “elastic strain limit” of concrete in compression:

1.8 cc

c

fE

ε ≈ (2.4)

Then apparent yielding of the member takes place at curvature:

1.8c cy

y c y

fd E d

εφξ ξ

= ≈ (2.5)

where the neutral axis depth at yielding, ξy (normalized to d) is still given by Eq. (2.2) with A, B:

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛ +++=⎟⎟

⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−++= 1

21

2'1

',11 2

2121w

v

wwscv

w bb

dt

bbB

bb

dt

bdEN

bbA

αδρ

δρραε

ρρρ (2.6)

The lower of the two values from Eqs. (2.1) or (2.5) is the yield curvature.

Equilibrium gives the yield moment:

( ) ( ) ( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦⎤

⎢⎣⎡ −+−+−

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ −+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

+= '1

6'1

2'1

221

21

32'1

2 21

2

3 δρρδξρξδξξδξ

ϕ Vyy

sy

wwyycy

y Edt

dt

dt

bb

bbE

bd

M (2.7)

If the outcome of Eq. (2.2) is less than the ratio of the flange thickness to the effective depth,

ξy<t/d, the neutral axis falls within the flange and the compression zone is rectangular. Then Eqs.

(2.3), (2.6), (2.7) should be applied anew with bw=b, degenerating into the expressions in [19] for

rectangular sections.

The criterion used for the identification of flexural yielding without adverse effects of shear is

the closeness of the experimental yield moment to the value from Eqs. (2.1)-(2.7), independently

of any parameter that has to do with shear effects. About 2500 specimens in the database with

rectangular, T-, H-, U- or hollow rectangular section and continuous longitudinal bars meet this

criterion. The test-to-predicted yield moment ratio plotted in Fig. 2.1 vs relevant parameters

suggests the following criteria for no flexure-shear coupling at yielding:

– Ls/h>3, or (2.8a)

– 2≤Ls/h≤3 and N<N1 or N>N2, or (2.8b)

– Ls/h<2 and 12

<sw

tot

Lh

ωω (2.8c)

where Ls is the shear-span (M/V-ratio), N the axial load, ωtot=ρtotfy/fc the total mechanical ratio of

7

longitudinal reinforcement and ωw=ρwfyw/fc that of transverse steel, and:

N1=0.5bhfc-As,totfy+ρwbwfyw[2Ls-(h-z)(ρtot-0.5ρv)/ρtot] (2.9)

N2=0.5bhfc+As,totfy-ρwbwfyw[2Ls+(h-z)(ρtot-0.5ρv)/ρtot] (2.10)

with z: internal lever arm. In the range of N between N1, N2 squat members with Ls/h<3 may

experience yielding of transverse steel and concrete diagonal compression failure before any of the

longitudinal reinforcement even yields.

Specimens meeting the criteria of Eqs. (2.8a) or (2.8b) are shown in Fig. 2.1(a) or 2.1(b),

respectively. Fig. 2.1(c) shows data violating only the 2nd condition of Eq. (2.8b); Fig. 2.1(d)

refers to very squat specimens meeting the 1st condition of Eq. (2.8c) and contrasts data meeting

its 2nd condition (i.e., with relatively light longitudinal reinforcement and heavy transverse one) to

those violating it. For about 2500 specimens in the database meeting Eqs. (2.8), rows 1-3 in Table

2.1 (at the end of Section 2) give statistics of the test-to-prediction ratio, while Fig. 2.2 compares

predictions to test data. Eq. (2.7) undershoots the experimental My because the latter is taken at the

corner of a bilinear force-deformation relation fitted to the measured response in primary loading

or to its envelope in cyclic. This point is slightly beyond yielding of the extreme tension steel or of

significant nonlinearity of the extreme compression fibres. Therefore, the “theoretical” yield

moment, My, and curvature, φy, from Eqs. (2.1)-(2.7) should be multiplied by 1.025 for beams/

columns, or 1.0 for walls, or 1.065 for members with a compression flange wider that the web.

The value of φy derived from the “experimental yield moment” by inverting Eq. (2.7) is

considered here as “experimental yield curvature”, φy,exp. Simple empirical expressions can be

fitted to φy,exp:

− for beams/columns: dEf

s

yy

154.1≈ϕ , or

hEf

s

yy

175.1≈ϕ (2.11a)

− for rectangular walls: dEf

s

yy

134.1≈ϕ , or

hEf

s

yy

144.1≈ϕ (2.11b)

− for T-, U-, H- or hollow rectangular sections: dEf

s

yy

147.1≈ϕ , or

hEf

s

yy

157.1≈ϕ (2.11c)

Being empirical, Eqs. (2.11) predict φy,exp with median test-to-prediction ratios of 1.0, but, as

8

they neglect important parameters, give larger coefficients of variation of test-to-predicted ratios

than Eqs. (2.1)-(2.6): 17.5%, 18.4%, 16.2% for the 1st version of Eq. (2.11a), (2.11b), (2.11c),

respectively, or 19.2%, 17.9%, 17.9% for the 2nd one.

In about 180 groups of specimens containing 2 to 9 (4 on average) nominally identical

specimens each, the intra-group variability of the measured My amounts to an average coefficient

of variation of 5%. This reflects test-to-test variability and the natural scatter of material properties

(e.g., of fy relative to means from few coupons, or of fc vis-à-vis the average from test cylinders) or

of geometric parameters (e.g., of d). The rest of the scatter is due to model uncertainty and

corresponds to coefficients of variation of the test-to-prediction ratio of My, φy, from Eqs. (2.1)-

(2.7) about equal to the values in Table 2.1 (at the end of this Section) reduced by 1%. For the

prediction of φy from Eqs. (2.11), it is about equal to the values quoted above minus 0.7%.

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0 1 2 3 4 5 6 7

My,

exp/M

y,p

red

(ωtot/ωw)/(2Ls/h)(a)

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0 5 10 15 20 25 30 35

My,

exp/M

y,p

red

(ωtot/ωw)/(2Ls/h)(b)

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0 5 10 15 20 25 30 35

My,

exp/

My,

pred

(ωtot/ωw)/(2Ls/h)

(c)

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0 5 10 15 20 25 30

My,

exp/M

y,p

red

(ωtot/ωw)/(2Ls/h)

`

(d) Fig. 2.1 Test-to-predicted yield moment vs (ωtot/ω w)/(2Ls/h) in members with continuous bars

and: (a) Ls/h>3; (b) 2≤Ls/h≤3 and N<N1 or N>N2; (c) 2≤Ls/h≤3 and N1<N<N2; (d) Ls/h<2

9

(a)

0

1000

2000

3000

4000

0 1000 2000 3000 4000

My,

exp

(kN

m)

My,pred (kNm)

median: My,exp=1.025My,pred

(b)

0

1500

3000

4500

6000

7500

0 1500 3000 4500 6000 7500

My,

exp

(kN

m)

My,pred (kNm)

median: My,exp= My,pred

(c)

0

2000

4000

6000

8000

10000

0 2000 4000 6000 8000 10000

My,

exp

(kN

m)

My,pred (kNm)


Fig. 2.2. Experimental yield moment v predicted from Eq. (2.7) in members with continuous bars:

(a) rectangular beams or columns; (b) rectangular walls; (c) members with T-, H-, U- or hollow

10

rectangular section

2.2 Fixed-end rotation due to pull-out of longitudinal bars from their anchorage beyond

the end section at uniaxial yielding

Curvature is generally measured as relative rotation of nearby sections divided by their distance.

At the end section of the member measured relative rotations often include the “fixed-end-

rotation” of the end section due to slippage of longitudinal bars from their anchorage beyond that

section, θslip. Rotations measured between the section of maximum moment and two different

nearby sections (i.e., with different gauge lengths) allow estimating the “fixed-end rotation” due to

bar slippage.

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

0 0.25 0.5 0.75 1 1.25 1.5lgauge / h

φy,

exp

/ (φ

y,1st

-pri

ncip

les

+φ y

,due

to s

lip )

Fig. 2.3. Yield curvature of members with continuous bars measured as relative rotation of two

sections including fixed-end rotation at the end section due to bar-pullout, divided by the sum of:

(a) the fixed-end rotation from Eq. (2.12) divided by the gauge length over which relative rotations

are measured, plus (b) the yield curvature from Eqs. (2.1)-(2.6).

The fixed-end rotation due to bar pull-out equals the slip from the anchorage zone divided by

the depth of the tension zone, (1-ξ)d. Assuming that bond stresses are uniform over a length lb of

the tension bars beyond the end section, the stress increases linearly along lb from 0 to the bar

elastic stress at the end section of the member, σs. Bar slippage from its anchorage equals

11

0.5σslb/Es. The ratio σs/Es to (1-ξ)d is φ. The length lb is proportional to the force in the bar, Asσs,

divided by its perimeter, πdbL (i.e. to 0.25dbLσs, where dbL is the mean tension bar diameter) and

inversely proportional to bond strength, i.e. to √fc. At yielding of the end section we have: φ=φy; if

we take the mean bond stress in MPa along lb equal to √fc(MPa) (which is about 50% or 40% of

the ultimate bond stress in unconfined or confined concrete, respectively) and set for simplicity

σs=fy (even when φy is obtained from Eq. (2.5)), the “fixed-end rotation” of the end section at

yielding is:

, 8y bL yl

y slipc

d ff

φθ = (fy, fc in MPa) (2.12)

Fig. 2.3 suggests that yield curvatures, φy,exp, measured as relative rotations of two sections,

often including the fixed-end rotation at the end section, agree well on average with the sum of:

(a) φy,slip =θy,slip/lgauge, with θy,slip from Eq. (2.12) and lgauge denoting the gauge length over which

relative rotations are measured, plus (b) φy from Eqs. (2.1)-(2.6). Besides, there is no systematic

effect of lg on this test-to-prediction ratio.

2.3 Chord rotation and secant stiffness to yield point in uniaxial flexure for members of

rectangular, T-, H-, U-, or hollow rectangular section

The overall deformation measure used here for a member is the chord rotation at each end, θ,

defined as the angle between the normal to the end section and the chord connecting the member

ends at the member’s displaced position.

Tension stiffening is small in members with longitudinal reinforcement ratio as high as that

typical of members designed for earthquake resistance. Moreover, the bond along bars between

cracks degrades with cyclic loading. So, as under seismic loading a member has normally been

subjected to one or more elastic load cycles by the time its end section yields, the small effect of

concrete in tension on the overall flexural deformations of the member at yielding is negligible.

So, the curvature may be taken to vary linearly along the shear span, Ls, contributing to the chord

12

rotation at yielding of the end section, θy, by φyLs/3. Diagonal cracking (assumed here at 45o)

starting at the end section spreads yielding of the tension reinforcement till the point where the

first diagonal crack intersects this reinforcement, i.e., up to a distance from the end equal to the

internal lever arm, z (with z=d-d’ in beams, columns, or members with T-, H-, U- or hollow

rectangular section, z=0.8h in rectangular walls). This increases the part of θy due to flexural

deformations from φyLs/3 to about φy(Ls+z)/3. The increase takes place only if flexural yielding at

the end section is preceded by diagonal cracking, i.e., if the shear force at diagonal cracking, VRc,

is less than that at flexural yielding of the end section, My/Ls.

The following expressions were fitted to the specimens meeting one of the criteria Eqs. (2.8):

− Rectangular beam/columns: slipysls

Vsyy a

LhzaL

,5.110014.03

θϕθ +⎟⎟⎠

⎞⎜⎜⎝

⎛++

+= (2.13a)

− Walls and members with hollow rectangular section:

slipyslVs

yy azaL

,0013.03

θϕθ +++

= (2.13b)

In the 3rd term of Eqs. (2.13) θy,slip is the “fixed-end rotation” at yielding from Eq. (2.12) and asl a

zero-one variable: asl=1 if slippage of longitudinal bars from their development beyond the end

section is possible, or asl=0 otherwise. In the 1st term φy is the theoretical yield curvature from Eqs.

(2.1)-(2.6) times the correction factor of 1.025, 1.0 or 1.065, for beams/columns, rectangular

walls, or members with T-, U-, H- or hollow rectangular section, respectively; av is a zero-one

variable: if VRc is the shear force at diagonal cracking (taken here equal to the shear resistance of

members without shear reinforcement in Eurocode 2 [12] and in the new fib Model Code [13]):

av=0, if VRc>My/Ls and av=1, otherwise. The 2nd term in Eqs. (2.13), attributed to shear

deformations along Ls, is purely empirical (from the fitting to the experimental data).

A fundamental simplification underlying the provisions of seismic design is that the global

inelastic response of the structure to monotonic lateral forces is bilinear, close to elastic-perfectly-

plastic. Then, the stiffness used in the elastic analysis should correspond to the stiffness of the

elastic branch of such a bilinear global force-deformation response. In a bilinear uniaxial force-

13

deformation (e.g., M-θ) model of a member that attributes all deformations to flexure of the shear

span, the member effective elastic rigidity may be taken as the secant stiffness to yield point:

y

syeff

LMEI

θ3= (2.14)

where My, θy are the moment and chord rotation, respectively, at the yielding end of the shear

span.

Eqs. (2.12)-(2.14) - with minor variations in some coefficients - were adopted in Part 3 of

Eurocode 8 [11].

When the amount and layout of longitudinal reinforcement is known (as in an assessment of

existing structures), the “theoretical effective stiffness” may be obtained by using in Eq. (2.14) the

values of My and θy from Eqs. (2.1)-(2.7) and (2.13), respectively. If the longitudinal

reinforcement has not been dimensioned yet (as in design of new structures), it is more convenient

to estimate the effective stiffness in terms of quantities already known at that stage. Statistical

analysis shows that such quantities affecting the ratio of the “experimental” EIeff to the uncracked

gross section stiffness, EIc, are: (a) the type of member (beam, column, wall, etc.); (b) the

possibility of slippage of bars from their anchorage beyond the end section; (c) the shear-span-to-

depth-ratio at the end of the member; and (d) the mean axial stress, N/Ac. The following “empirical

effective stiffness” has been fitted directly to test results:

( ) ceff EI50;min048.016.0;maxln8.025.01EI ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛+−= MPa

AN

hL

aac

sslEI (2.15)

where asl is the zero-one variable defined for the 3rd term of Eqs. (2.13) and:

– aEI=0.108 for columns;

– aEI=0.133 for beams;

– aEI=0.165 for rectangular walls.

– aEI=0.118 for members with T-, H-, U- or hollow rectangular section.

For the various types of specimens meeting the criteria of Eqs. (2.8) for flexural yielding

without effects of shear, rows 4-8 in Table 2.1 (at the end of this Section) give statistics of the test-

14

to-prediction ratio for the chord rotation at yielding and rows 9-14 of the ratio of the experimental

secant stiffness to yielding to the values from Eqs. (2.14) or (2.15). Natural and test-to-test

variability contribute to the scatter with a coefficient of variation of about 10% in practically

identical specimens. The rest of the scatter, due to model uncertainty, corresponds to coefficients

of variation equal to the values in Table 2.1 minus about 1.5%.

rectangular beams and columns rectangular walls members with T-, H-, U- or hollow rectangular section

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3 3.5

θ y,e

xp(%

)

θy,pred (%)

median: θy,exp=1.01θy,pred

(a)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

θ y,e

xp(%

)

θy,pred (%)


(b)

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 1.4

θ y,e

xp(%

)

θy,pred (%)


(c)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

(MyL

s/3θ y

) exp

(MNm

2 )

(MyLs/3θy)pred (MNm2)

median: (MyLs/3θy)exp =1.01(MyLs/3θy)pred

(d)

0

300

600

900

1200

1500

0 300 600 900 1200 1500

(MyL

s/3θ

y)exp(M

Nm

2 )


median: (MyLs/3θy)exp=1.025(MyLs/3θy)pred

(e)

0

1500

3000

4500

6000

7500

9000

0 1500 3000 4500 6000 7500 9000

(MyL

s/3θ

y)ex

p (M

Nm

2 )


median:(MyLs/3θy)exp=0.995(MyLs/3θy)

(f)

0

100

200

300

400

500

600

0 100 200 300 400 500 600

(MyL

s/3θ y

) exp

(MNm

2 )

EIpred (MNm2)

median: (MyLs/3θy)exp =1.01EIpred

(g)

0

300

600

900

1200

1500

0 300 600 900 1200 1500

(MyL

s/3θ

y)exp(M

Nm

2 )

EIpred (MNm2)

median: (MyLs/3θy)exp=0.995EIpred

(h)

0

1500

3000

4500

6000

7500

9000

0 1500 3000 4500 6000 7500 9000

(MyL

s/3θ

y)ex

p (M

Nm

2 )

EIpred (MNm2)

median:(MyLs/3θy)exp=EIpred

(i) Fig. 2.4 Experimental vs. predicted chord rotation or secant stiffness at flexural yielding of

members with continuous bars: (a)-(c): chord rotation at yielding vs. Eqs. (2.13); (d)-(f): secant

stiffness at yielding vs. Eq. (2.14); (g)-(i): secant stiffness to yield point vs Eq. (2.15)

15

Fig. 2.4 compares experimental to predicted chord rotations or secant stiffness at yield point

for the three major member groupings considered here. Witness in Figs. 2.4(a) and (b) the inability

of Eqs. (2.13a), (2.13b) to explain extreme (very low or very large) values of θy for rectangular

beams, columns or walls and in Figs. 4(d) and (e) the downwards bias of Eq. (2.14) for the stiffest

among these types of elements. Table 2.1 (at the end of Section 2) shows that Eq. (2.15), being

purely empirical, provides a better mean fit to the data than Eq. (2.14). However, for other than

rectangular walls Eq. (2.15) has larger prediction scatter, as it neglects the effect of longitudinal

reinforcement. Moreover, its scope is limited to the range of parameter values in the database.

About 50 tests on members with continuous ribbed longitudinal bars and about 30 tests with

smooth ones, all without seismic detailing (i.e., with sparse ties without 135o hooks) show that My,

θy and EIeff are not affected by poor detailing and can still be described by the models above.

2.4 Yield moment, chord rotation and secant stiffness to yield point under biaxial flexure

If yielding under biaxial loading is identified with the corner of a bilinear moment-chord rotation

envelope of the measured hysteresis loops in each direction of bending, the available bidirectional

tests on columns or walls suggest the following (see also rows 1-7 from the bottom in Table 2.1).

1. The “experimental yield moments” in the two directions of bending agree well on average

with the components of flexural resistance under biaxial loading computed from plane-section

analysis, elastic-perfectly plastic steel and parabolic σ-ε law for concrete up to fc and εco=0.002

and horizontal thereafter, until a compressive strain of 0.0045 at one corner of the section.

2. With the uniaxial chord rotations at yielding, θyy,uni, θyz,uni, obtained from Eqs. (2.13), the

“experimental” chord rotation components at yielding, θyy,exp, θyz,exp, fall, by about 11% on

average, outside the interaction diagram:

2 2

,exp ,exp

, ,

1yy yz

yy uni yz uni

⎛ ⎞ ⎛ ⎞θ θ+ =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟θ θ⎝ ⎠ ⎝ ⎠

(2.16)

3. Owing to 2 above, the “experimental” secant stiffness to yield point in each direction of

16

biaxial bending is on average about 11% less than the uniaxial “theoretical effective stiffness”

of Eq. (2.14). It exceeds also by about 7% the empirical stiffness from Eq. (2.15). These

differences are small and – in view of their limited experimental support – do not seem worth

taking into account in analysis.

Table 2.1: Statistics of test-to-predicted yield properties – members with continuous bars

Test-to-prediction ratio No of data

mean* median* coefficient of variation

1 My,exp/My,Eq. (2.7) beams/columns - uniaxial loading 2084 1.045 1.025 16.3% 2 My,exp/My,Eq. (2.7) rectangular walls - uniaxial loading 175 0.995 0.995 15.5% 3 My,exp/My, Eq. (2.7) uniaxial T-, H-, U- or hollow rectangular sections 167 1.08 1.065 12.4% 4 θy,exp/θy,Eq. (2.13a) beams/columns – uniaxial tests without bar slip 284 1.07 1.035 26.3% 5 θy,exp/θy,Eq. (2.13a) beams/columns - uniaxial tests with bar slip 1368 1.045 1.00 33.2% 6 θy,exp/θy,Eq. (2.13a) beams/columns – all uniaxial tests 1652 1.05 1.01 32.0% 7 θy,exp/θy,Eq. (2.13b) rectangular walls (uniaxial, with bar slip) 164 0.99 0.96 37.9% 8 θy,exp/θy,Eq. (2.13b) T, H, U, hollow rectang. sections (uniaxial w/ bar slip) 152 1.35 1.005 29.4% 9 θy,exp/θy,Eq. (2.13b) walls and hollow piers 316 1.02 0.975 33.2%

10 (MyLs/3θy)exp/(MyLs/3θy)Eq. (2.14) beams/columns - uniaxial loading 1615 1.06 1.01 32.1% 11 (MyLs/3θy)exp/EIeff Eq. (2.15) beams/columns - uniaxial loading 1615 1.06 1.01 35.9% 12 (MyLs/3θy)exp/(MyLs/3θy)Eq. (2.14) rectangular walls - uniaxial loading 164 1.16 1.025 51.3% 13 (MyLs/3θy)exp/EIeff Eq. (2.15) rectangular walls - uniaxial loading 164 1.065 0.995 49.7% 14 (MyLs/3θy)exp/(MyLs/3θy)Eq. (2.14) T, H, U, hollow rect. sections - uniaxial 152 1.05 0.995 35.3% 15 (MyLs/3θy)exp/EIeff Eq. (2.15) T, H, U, hollow rectang. sections – uniaxial 152 1.09 1.00 42.3% 16 (MyLs/3θy)exp/(MyLs/3θy)Eq. (2.14) walls and hollow piers 316 1.105 1.01 44.0% 17 (MyLs/3θy)exp/EIeff Eq. (2.15) walls and hollow piers 316 1.075 1.00 45.1% 18 Myy,exp/Myy,pred.-1st-principles - biaxial loading 35 1.00 0.99 11.6% 19 Myz,exp/Myz,pred.-1st-principles - biaxial loading 35 1.00 0.98 12.0% 20 SRSS of θyy,exp/θyy,Eq. (2.13) & θyz,exp/θyz,Eq. (2.13) - biaxial loading 34 1.16 1.11 21.9% 21 (MyyLs/3θyy)exp/EIeff,y-Eq. (2.14) - biaxial loading 34 0.93 0.90 24.4% 22 (MyzLs/3θyz)exp/EIeff,z-Eq. (2.14) - biaxial loading 34 0.93 0.87 23.9% 23 (MyyLs/3θyy)exp/EIeff,y-Eq. (2.15) - biaxial loading 34 1.05 1.07 23.4% 24 (MyzLs/3θyz)exp/EIeff,z-Eq. (2.15) - biaxial loading 34 1.08 1.07 26.2% * If the sample size is large, the median is more representative of the average trend than the mean, for instance, the median of the predicted-to-test ratio is always the inverse of the median of the test-to-predicted ratio, whereas the mean of both ratios typically exceeds their median.

17

3 Effect of lap-splicing of longitudinal bars in the plastic hinge zone on

member flexure-controlled yielding and secant stiffness to yield point 3.1 Yield moment and curvature of section with rectangular, T- or L- compression zone

Beams/columns, rectangular walls or members with T-, H-, U- or hollow rectangular section and

continuous bars meeting the criteria of Eqs. (2.8) were found above to have an “experimental” My

that exceeds by an average factor of 1.025, 1.0, or 1.065, respectively, the value from Eqs. (2.1)-

(2.7). In Figs. 5(a), 6(a), which refer to members with longitudinal bars lap-spliced starting at the

end section, the points to the right of 1.0 on the horizontal axis have relatively long lap lengths.

The corresponding “experimental” My exceeds the value predicted considering the bars as

continuous by an average factor greater than the factors quoted above for members with

continuous bars. To make up for (at least part) of the difference, it is proposed to increase the

computed yield moment by including both bars in any pair of lapped compression bars as

compression reinforcement. In other words, end bearing of a compression bar stopping at the end

section against the very well confined concrete beyond that section is considered sufficient for the

build-up of a compressive stress in the bar almost as high as in its companion in the lap which

continues past the end section. Compatibility of longitudinal strains between these two bars and

the concrete surrounding them near the member’s end section contributes to this effect. As shown

in Figs. 5(b), 6(b), the “experimental” My of the points to the right of the value of 1.0 on the

horizontal axis compares then better with the value from Eqs. (2.1)-(2.7).

For the lapped tension bars, the maximum stress that one of them can develop, fsm, may be

taken from the model in [10], fitted to over 800 tests for anchorage or lap-splicing of ribbed bars

and adopted in the new fib Model Code [13]. According to it, for “good” bond conditions the

expected value of fsm at distance lo from its end is about the same in a single anchored bar or in one

lap-spliced with a parallel bar placed at a clear distance not exceeding 4dbL:

yo

trdbL

d

bL

bo

co

c

bL

o

smo

sm fppkK

cc

dc

dd

ff

dl

ff

≤⎥⎥

⎦

⎤

⎢⎢

⎣

⎡++⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

1.0max

3/12.025.055.0

;1min (3.1)

18

Where [10]:

− fy : yield stress of the bar;

− fsmo=51.2MPa;

− dbL: bar diameter;

− fco=20MPa;

− dbo: 20mm;

− cd=min [minc; a/2], limited between dbL and 3dbL;

− cmax=max [maxc; a/2], with an upper limit of 5cd, where:

• minc and maxc: the minimum and the maximum, respectively, clear cover of the bars,

• a: clear distance between anchored bars or pairs of lapped bars;

− 1 0.04l shtr

b bL h

n AKn d s

= ≤ (3.2)

is the total cross-sectional area of reinforcement placed within length lo transverse to the axis

of the bar(s) and intersecting the potential splitting crack, divided by dbLlo; in Eq. (3.2):

• nb: number of anchored bars or pairs of lapped bars within the plane of the potential

splitting crack, from the bar(s) to the concrete surface,

• nlAsh/sh: total cross-sectional area of legs of transverse reinforcement crossing the splitting

crack, per unit length of the lapped or developed bar;

− k: effectiveness factor, equal to k=0, except for:

• k=10, if the transverse reinforcement is at right angles to the splitting plane;

• k=5, if the potential splitting extends from the bar perpendicular to the surface and is

crossed at right angles by a transverse reinforcement placed within the cover (for clear

distance between developed bars or pairs of lapped bars less than three-times the cover);

− p: “active” confining pressure normal to the axis of the developed or lapped bars, due to

external loading;

− po=5MPa.

19

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5

My,

exp

/My,

pre

d

fsm/fy

beams&columns

T,H,U or hollow rect.

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5

My,

exp

/My,

pre

d

fsm/fy

beams&columns


(b)

Fig. 3.1 Ratio of experimental My of members with lap-spliced bars to My predicted: (a) neglecting

the lap splicing; (b) considering the lap splices by counting fully both spliced compression bars

and using the steel stress from [10], [13] for the tension bars

Rows 1-3 in Table 3.1 (at the end of Section 3) list statistics of the ratio of the experimental My

to the value predicted taking into account the lap-splicing according to the above rules based on

[10], [13]. The median value appears to be satisfactory, but this is thanks to the data where fsm

according to [10], [13] is equal to fy. Fig. 3.1 shows how:

– the ratio of experimental My to the value predicted neglecting the lap splicing (Fig. 3.1(a)), and

– the ratio of the experimental My to the one predicted taking into account the lap splicing

according to the rules above (Fig. 3.1(b))

vary with the ratio of fsm according to [10], [13] to fy. The improvement in the prediction in Fig.

3.1(b) compared to Fig. 3.1(a) seems insufficient for short lappings and low values of fsm.

A simpler alternative to the use of fsm according to [10], [13], adopted already in Part 3 of

Eurocode 8 [11], is the following:

yoy

osm f

llf ⎟

⎟⎠

⎞⎜⎜⎝

⎛=

min,

,1min (3.3)

where:

,min 0.3 /oy bL y cl d f f= (fy, fc in MPa) (3.4)

Eq. (3.4) implies that the lapped bars develop a uniform bond stress along the lapping equal to

20

√fc/1.2 (MPa).

Eq. (3.3) has been proposed in [1], using though in lieu of loy,min the bar development length

according to ACI-318, which is in general longer than that from Eq. (3.4).

The statistics of the test-to-prediction ratio of My are listed in rows 4-6 of Table 3.1 (at the end

of Section 3), this time using Eqs. (3.3), (3.4), instead of Eq. (3.1). The ratio of experimental My to

the one predicted, first neglecting the lap splicing, Fig. 3.2(a), and then taking it into account

according to Eqs. (3.3), (3.4), Fig. 3.2(b), is plotted in Fig. 3.2 vs lo/loy,min. There is a marked

improvement in the prediction for low values of lo and My.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5

My,

exp/

My,

pred

lo/loy,min

beams&columns


(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5

My,

exp/

My,

pred

lo/loy,min

beams&columns


(b) Fig. 3.2 Ratio of experimental My of members with lap-spliced bars to My predicted: (a) neglecting

the lap splicing; (b) considering the lap splices by counting fully both spliced compression bars

and using Eq. (3.3) for the tension bars

3.2 Chord rotation and effective stiffness at yielding

It is proposed here to calculate θy still from Eqs. (2.13), but using in the 1st and the 3rd terms of the

right-hand-side a value of φy consistent with that of My: with all bars in any pair of lapped

compression bars counted as compression reinforcement and the yield stress of lapped tension bars

reduced according either to Eqs. (3.1), (3.2) or to Eqs. (3.3), (3.4). Besides, the 2nd term in Eqs.

(2.13) is multiplied by the ratio of the My modified for the lap splicing, to the value of My outside

the lap splice. Moreover, to determine whether we have aV=1 in the 1st term of Eqs. (2.13), the

value of the end moment at the time diagonal cracking takes place, LsVRc, is compared to the a

21

value of My that accounts for the lap splice.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5

θ y,e

xp/θ

y,p

red

lo/loy,min

beams&columns


(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5

θ y,e

xp/θ

y,pr

ed

lo/loy,min

beams&columns


(b) Fig. 3.3. Ratio of experimental chord rotation at yielding of members with lap-spliced bars to one

predicted: (a) neglecting the lap splicing; (b) considering the lap splices by counting fully both

spliced compression bars and using Eq. (3.3) for the tension bars

The statistics of the test-to-prediction ratio of θy are listed in rows 7-9 or 10-12 of Table 3.1 (at

the end of Section 3), if Eqs. (3.1), (3.2) or Eqs. (3.3), (3.4), respectively, are used for the lapped

tension bars. Rows 13-15 and 16-18 in Table 3.1 list the corresponding statistics for the effective

stiffness from Eq. (2.14). The ratio of the experimental θy to the one predicted is plotted vs

lo/loy,min in:

– Fig. 3.3(a), neglecting the lap splicing, and

– Fig. 3.3(b), taking it into account according to the rules above and using Eqs. (3.3), (3.4) for

the lapped tension bars.

Fig. 3.4 makes a similar presentation for the secant stiffness to yield point, first neglecting (Fig.

3.4(a)) and then considering (Fig. 3.4(b)) the effect of lap splicing as proposed above.

Interestingly, the secant stiffness to yield point in Fig. 3.4(a) computed neglecting the lap splice

compares at least as well with the experimental data as the value in Fig. 3.4(b) which considers it.

So, lap splicing does not seem to reduce the secant stiffness to yielding.

The coefficients of variation of the test-to-prediction ratios of all yield properties of members

with lap splices are smaller than their counterparts for members with continuous bars in Table 2.1

22

(at the end of Section 2), but this is mostly due to the much fewer data.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5

EI e

xp/E

I pre

d

lo/loy,min

beams&columns


(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5

EI ex

p/EI pr

ed

lo/loy,min

beams&columns


(b) Fig. 3.4. Ratio of experimental stiffness to yield point of members with lap-spliced bars to value

from Eq. (2.14): (a) neglecting the lap splicing; (b) considering the lap splices by counting fully

both spliced compression bars and using Eq. (3.3) for the tension bars, etc.

Table 3.1: Statistics of test-to-predicted yield properties - members with lap-spliced bars

Test-to-prediction ratio No of data


1 My,exp/My,1st principles rectangular columns – steel stress from Eqs. (3.1), (3.2) 103 0.99 0.98 12.2%

2 My,exp/My,1st principles rect. walls, T-, H-, U-, hollow rect. sections – steel stress from Eqs. (3.1), (3.2) 11 0.98 0.985 8.9%

3 My,exp/My,1st principles all members – steel stress from Eqs. (3.1), (3.2) 114 0.99 0.98 12.0%

4 My,exp/My,1st principles rectangular columns – steel stress from Eqs. (3.3), (3.4) 103 1.00 0.985 12.0% 5 My,exp/My,1st principles rect. walls, T, H, U, hollow rect. sections–Eqs.(3.3)-(3.4) 11 1.02 1.02 10.3% 6 My,exp/My,1st principles all members – steel stress from Eqs. (3.3), (3.4) 114 1.00 0.995 11.8%

7 θy,exp/θy,pred rectangular columns – steel stress from Eqs. (3.1), (3.2) 81 1.05 1.035 19.8%

8 θy,exp/θy,pred rect. walls, T-, H-, U-, hollow rect. sections – steel stress from Eqs. (3.1), (3.2) 11 0.965 0.98 25.7%

9 θy,exp/θy,pred all members – steel stress from Eqs. (3.1), (3.2) 92 1.04 1.03 20.5%

10 θy,exp/θy,pred rectangular columns – steel stress from Eqs. (3.3), (3.4) 81 1.06 1.045 19.1% 11 θy,exp/θy,pred rect. walls, T-, H-, U-, hollow rect. sections – Eqs. (3.3), (3.4) 11 1.025 1.08 27.2% 12 θy,exp/θy,pred all members – steel stress from Eqs. (3.3), (3.4) 92 1.055 1.05 20.0%

13 (MyLs/3θy)exp/(MyLs/3θy)pred rect. columns–steel stress from Eqs. (3.1), (3.2) 81 0.985 0.96 24.6%

14 (MyLs/3θy)exp/(MyLs/3θy)pred rect. walls, non-rectangular sections – steel stress from Eqs. (3.1), (3.2) 11 1.07 1.00 24.7%

15 (MyLs/3θy)exp/(MyLs/3θy)pred all members – steel stress from Eqs. (3.1), (3.2) 92 0.99 0.97 24.6%

16 (MyLs/3θy)exp/(MyLs/3θy)pred rect. columns–steel stress from Eqs. (3.3), (3.4) 81 0.98 0.955 24.8% 17 (MyLs/3θy)exp/(MyLs/3θy)pred rect. walls, non-rect. sections – Eqs. (3.3), (3.4) 11 1.055 1.00 25.7% 18 (MyLs/3θy)exp/(MyLs/3θy)pred all members – steel stress from Eqs. (3.3), (3.4) 92 0.99 0.955 24.8%

* See footnote of Table 2.1 (at the end of Section 2).

23

4 Flexure-controlled ultimate deformations of members with continuous bars

4.1 Uniaxial ultimate curvature of members with rectangular compression zone

The ultimate deformation of a RC section or member is conventionally identified with the post-

ultimate strength point of the lateral load-deformation response where any increase in the imposed

deformations cannot increase the resisting force above 80% of ultimate strength. The so-defined

ultimate curvature, φu, is calculated here explicitly without a full moment-curvature analysis.

Plane section analysis is carried out with the following nonlinear σ-ε laws:

- For the concrete, the σ-ε law rises as a parabola until a strain εco and stays then horizontal up to

a strain less or equal to the ultimate strain, εcu. In the core of the section inside the stirrups the

values of εco, εcu and of the concrete strength increase thanks to confinement.

- At the relatively low steel strains associated with section ultimate conditions due to concrete

crushing, reinforcing steel is taken as elastic-perfectly plastic. At the large strains typical of

failure due to steel rupture the steel is taken as elastic-linearly strain-hardening, from the yield

stress fy at the strain at the outset of strain-hardening, εsh, to the ultimate strength ft at

elongation εsu. The tension- the compression- and the web-reinforcement in-between may have

different post-elastic properties, indexed by 1, 2 or v, respectively.

If failure is due to steel rupture at elongation εsu before concrete crushing φu is:

( )dsu

susu ξ

εϕ−

=1

(4.1)

where ξsu is the neutral axis depth at such a failure (normalized to d). If the section fails by

crushing of the extreme concrete fibres while the neutral axis depth is ξcu (normalized to d), the

ultimate curvature is:

dcu

cucu ξ

εϕ = (4.2)

Flow Chart 1 shows how ξsu, ξcu are determined from Eqs. (4.8)-(4.10) below depending on

the value of the distance of the extreme tension or compression reinforcement to the concrete

24

surface, d’ (normalized to d as δ’ =d’/d) with respect to the limits of Eqs. (4.3), (4.4) and of the

axial-load-ratio ν=N/bdfc relative to those of Eqs. (4.5)-(4.7). In these expressions: ω1=ρ1fy1/fc,

ω2=ρ2fy2/fc, ωv=ρvfyv/fc, with ρ1, ρ2, ρv normalized to bd.

Flow Chart 1: Calculation of ultimate curvature for the full section before or at spalling of

concrete cover

Unconfined full section – Steel rupture

δ' satisfies Eq. (4.3)?

no

Unconfined full section – Spalling of concrete cover

δ' satisfies Eq. (4.4)?

yes

no

ξsu from Eq. (4.9), MRc from Eq. (4.14)

ν<νs,y2 - LHS Eq. (4.5)?

ν<νs,c - RHS Eq. (4.5)?

yes

ν<νs,c - RHS Eq. (4.5)?

yes ξsu from Eq. (4.8), MRc from

Eq. (4.15)

φsu from Eq. (4.1)

no

ν<νc,y2 - LHS Eq. (4.6)?

ν<νc,y1 - RHS Eq. (4.6)?

ξcu from Eq. (4.12), MRc, from Eq. (4.16)

ξcu from Eq. (4.11), MRc from Eq.

(4.18)

yes

ξcu from Eq. (4.10), MRc from Eq. (4.17)

no

no

no

Compute moment resistances: − MRc (of full, unspalled section) and − MRo (of confined core, after spalling of cover).

ν<ν c,y2 - RHS Eq. (4.7)?

ν<ν c,y1 - LHS Eq. (4.7)?

yes

no

no

ξcu from Eq. (4.13), MRc from Eq. (4.19)

MRo< 0.8MRc?φcu from Eq. (4.2) yes

Ultimate curvature of confined core after spalling of concrete cover

yes yes

no

Flow Chart 2

yes

yes

no

25

Flow Chart 2: Calculation of ultimate curvature for the confined core of the section after spalling

of concrete cover

sucu

ycu

εε

εεδ

+

−≤ 2' (4.3)

1

2'ycu

ycu

εε

εεδ

+

−≤ (4.4)

( )( )

( )( ) ( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+−−+

+−−−+

+

−

≡≤≤≡⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+−

+−−+

+

−−+

yv

tvshvsucusucusu

cusu

v

y

t

sucu

cocu

csysyv

tvshvsuysu

ysu

v

y

t

ysu

coysu

ff

ff

ff

ff

121'

'13

1213

'1'

11111

112

1

,2,121211

112

21

2

εεεεεεδεεδ

ωωωεε

εε

νννεεεεεε

ωωωεε

εδεεδ

(4.5)

Rupture of tension steel

Failure of compression zone (concrete)

no

ξ*su from Eq. (4.9), MRo from Eq.

(4 14)

ν*<ν*s,y2 – LHS Eq. (4.5)?

ν*<ν*s,c – RHS Eq. (4.5)?

yes ξ*su from Eq. (4.8), MRo from Eq.

(4 15)

φsu from Eq. (4.1)

ν*<ν*c,y2 - LHS Eq.

ξ*cu from Eq. (4.12), MRo

from Eq. (4.16)

yes

ξ*cu from Eq. (4.10), MRo from Eq. (4.17)

no

no

no

Confined core after spalling of concrete cover. Parameters are denoted by an asterisk and are computed with: − b, d, d’ replaced by geometric parameters of the core: bo, do, d’o; − N, ρ1, ρ2, ρv normalized to bodo, instead of bd; − σ-ε parameters of confined concrete, fcc, εcc, used in lieu of fc, εcu

ν*<ν*c,y1 - RHS Eq.

ξ*cu from Eq. (4.11), MRo from Eq.

(4 18)yes

φcu from Eq. (4.2)

yes

26

11

1

112

1,2,22

212

3'1

3'1''1

ycu

cocu

ycu

ycuv

ycycycu

cocu

ycu

ycuv

εε

εε

δεεεε

δω

ωω

νννεε

εε

δεεεε

δδ

ωωω

+

−+⎟

⎟⎠

⎞⎜⎜⎝

⎛−

+

−

−+−

≡<≤≡−

−+⎟

⎟⎠

⎞⎜⎜⎝

⎛−

−

+

−+−

(4.6)

( )

22

2

1

12

2,1,1

112

2

3''1'1

'')'1(

3'1'1

2')'1(

ycu

cocu

cuyyv

vycu

y

ycycycu

cocu

cuyv

vycu

y

εε

εε

δεεδδ

εδω

δεεδ

εω

ω

νννεε

εε

δδε

εω

ωεδεδεω

−

−+⎟

⎠⎞

⎜⎝⎛ −

−+

+−−

−

≡<≤≡+

−+⎟

⎠⎞

⎜⎝⎛

−+

−+−−− (4.7)

( )

( ) vyv

tv

su

shv

su

co

vyv

tv

su

shv

su

co

y

t

su

ff

ff

ff

ωεε

εε

δ

ωεε

δεε

ωωνδ

ξ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−++⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+++⎟

⎟⎠

⎞⎜⎜⎝

⎛+−+−

≈

11212

31'1

1121'1

3'1

11

12

1

11

(4.8)

0'3

11)'1(2

'3

'3

11)'1(3

21

311

)'1(231

12

11221

1

112

12

1

11

21

11

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

−++++

+⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

−+++++

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

−++

yv

su

su

yvshv

su

shv

yv

tv

y

su

yv

tv

su

co

yv

su

su

yvshv

su

shv

yv

tv

y

su

y

t

su

co

yv

su

su

yvshv

su

shv

yv

tvv

su

co

ff

ff

ff

ff

ff

εεδ

εεε

εε

δω

εεδωω

εεν

ξεεδ

εεε

εε

δω

εεωω

εεν

ξεε

εεε

εε

δω

εε

ν

ν (4.9)

( )( ) ( )

( ) vcu

co

vcu

ωεε

δ

ωδωωνδξ

23

1'1

'1'1 21

+⎟⎟⎠

⎞⎜⎜⎝

⎛−−

++−+−= (4.10)

( )( ) 0

'12'1)'1(231

1

11

112

22

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+−++

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −

−−− cu

yv

v

yyv

cu

y

cu

yvcu

yvcuv

cu

co εεδ

ωεω

ξδεε

δω

νεε

ωωξεε

εε

δω

εε ν (4.11)

0)'1(2

)'1(')'1(3

12

2

1

1

2

2

1

12 =⎟⎟⎠

⎞⎜⎜⎝

⎛

−+

++−⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

−++−−⎥

⎦

⎤⎢⎣

⎡− cu

yvyycu

yvyycu

co εεδδω

εωδ

εωξε

εδω

εω

εωνξ

εε νν (4.12)

( )( ) 0'

'12''1

'1)'1(231

2

2

221

22

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+−+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +

−+− δε

εδδω

εωξ

εδε

δω

εεωωνξ

εεεε

δω

εε ν

cuyv

v

yyv

cu

y

cu

yvcu

yvcuv

cu

co (4.13)

If the concrete cover spalls before the tension reinforcement ruptures and the section has a

well-confined concrete core, the moment resistance of the confined core, ΜRo, may eventually

increase above 80% of the moment resistance of the unconfined full section, ΜRc. Then the section

27

reaches the ultimate condition in flexure after spalling. Flow Chart 2 applies then, with the

confined core taken as the section (its dimensions denoted by an asterisk) and with the properties

of the confined concrete, fcc, εcc, used in lieu of fc, εcu. (see next section for the values of these

properties). ΜRc and ΜRo may be computed from Eqs. (4.14)-(4.19) according to Flow Charts 1

and 2.

( ) ( )

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

−+⎥

⎦

⎤⎢⎣

⎡−−

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+−

−

+⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

+−

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+−−−=

)1(11)1(13

)'1(2)1(2

'11

'11'1)'1(6

1'

2)'1(1

421

321

111

1

1

2

12

1

11

112

ξεεξ

εεδ

εε

ξδξδξ

εε

εε

ξδδ

ω

εε

ξδξωωδξ

εεξ

εεξξ

ν

yv

tv

su

shv

su

shv

su

yv

yv

su

su

yv

y

su

y

t

su

co

su

co

c

R

ff

ff

fbdM

(4.14)

( ) ( )

( )⎪⎩

⎪⎨⎧

⎪⎭

⎪⎬⎫

−⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−+⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−−−

−+

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+−−−=

)1(116

114

)'1(1

31)1)('(

'1

2)'1(

142

132

1

1

1

11

2

1

21

11

112

ξεεξ

εεδ

ε

εξξδξ

δω

ωωδ

ξεε

ξεεξξ

ν

y

t

su

shv

su

shv

su

yv

y

t

su

co

su

co

c

R

ff

ff

fbd

M

(4.15)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛+

−

+⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−

−=

cu

yv

yv

cu

cu

yv

y

cu

cu

co

cu

co

c

R

fbdM

ε

εξ

δξδξ

εε

δε

εξ

δω

εε

ξδξ

ωωδ

ξεε

ξεεξξ

ν 132

3111

)1(4

2)1(

421

321

111

1

2

121

12

(4.16)

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

−+

+−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−

−=

2

1

212 3

1)1)('(12

))('1(42

132

1

cu

yv

cu

co

cu

co

c

R

fbdM

ε

ξεξδξ

δωωωδ

ξεε

ξεεξξ ν (4.17)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−

−=

cu

yv

yv

cu

cu

yv

y

cu

cu

co

cu

co

c

R

fbdM

ε

εξδ

ξξ

εε

ε

εξ

δω

ωεε

ξξω

δξ

εε

ξεεξξ

ν 132'

311111

)'1(4

12

)'1(42

132

12

112

(4.18)

yv

cu

yy

cu

cu

co

cu

co

c

R

fbdM

εε

ξδω

εω

δξεω

ξξεδ

ξεε

ξεεξξ ν

12)'1(

)'()1(2

)'1(42

132

1 2

2

2

1

12

−+⎟

⎟⎠

⎞⎜⎜⎝

⎛−+−

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−

−= (4.19)

4.2 Strains of steel and concrete at section ultimate curvature

The ultimate curvatures, φu, calculated according to the previous section fits best test results at

flexure-controlled failure of members with rectangular compression zone if the material

parameters are chosen as follows:

28

The confined concrete strength, fcc, is taken according to the model in [18], which has been

adopted in Part 3 of Eurocode 8 [11]:

)1( Kff ccc += , 86.0

7.3 ⎟⎟⎠

⎞⎜⎜⎝

⎛≈

c

yws

ff

Kαρ

(4.20)

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 [21]:

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−= ∑

oo

i

o

h

o

h

hbb

hs

bs 6/

12

12

12

α (4.21)

with sh denoting the centreline spacing of stirrups, bo and ho the confined core dimensions to the

centreline of the hoop and bi the centreline spacing along the section perimeter of longitudinal bars

(indexed by i) engaged by a stirrup corner or a cross-tie. The strain at fcc is taken as [20]:

)51( Kcocc += εε (4.22)

with εco =0.002 and K from Eq. (4.20). For fcc and εcc according to these models, the free

parameters to fit the experimental values of φu are the ultimate strains of steel, εsu, and of confined

or unconfined concrete, εcu,c, all taken different for monotonic or cyclic loading.

00.0020.0040.0060.0080.01

0.0120.0140.0160.018

0 100 200 300 400 500 600 700 800 900

ε cu,

'exp

'

h (mm)

00.0020.0040.0060.0080.01

0.0120.0140.0160.018

0 25 50 75 100 125 150 175 200 225 250 275 300

ε cu,

'exp

'

ξyd (mm) Fig. 4.1 Strain in extreme compression fibre corresponding to the experimental spalling curvature

of the section, compared to the predictions of: (a) Eq. (4.23); (b) Eq. (4.24)

The value of εcu,c that provides the best fit to the data is shown in Fig. 4.1 and suggests a size

effect. Such an effect has indeed been observed experimentally in [5]. If ho, do, ξdo are the depth,

effective depth and neutral axis depth, respectively, of the confined core (taken equal to those of

29

the full section for spalling of the extreme compression fibres) and ρw is the transverse steel ratio

in the direction of bending, the value of εcu,c providing the best fit to the data is:

- for cyclic loading: cc

yww

occu

f

fmmh

αρε 4.0

)(100035.0

2

, +⎟⎟⎠

⎞⎜⎜⎝

⎛+= , or (4.23a)

cc

yww

occu

f

fmmd

αρξ

ε 4.0)(

10035.02/3

, +⎟⎟⎠

⎞⎜⎜⎝

⎛+= , or (4.24a)

- for monotonic loading: cc

yww

occu

f

fmmh

αρε 57.0

)(100035.0

2

, +⎟⎟⎠

⎞⎜⎜⎝

⎛+= , or (4.23b)

cc

yww

occu

f

fmmd

αρξ

ε 57.0)(

10035.02/3

, +⎟⎟⎠

⎞⎜⎜⎝

⎛+= (4.24b)

The limits on εsu that provide the best fit to the data are:

– for cyclic loading: alnosucysu min,, 83εε = (4.25)

– for monotonic loading: alnosumonsu min,, 127 εε = (4.26a)

or more generally: alnosutensionbmonsu N min,,, ln311 εε ⎟

⎠⎞

⎜⎝⎛ −= (4.26b)

where εsu,nominal is the uniform elongation at tensile strength in the standard test of steel coupons

and Nb,tension the number of bars in the tension zone. The adverse effect of load cycling on steel

bars (surface cracking upon buckling, etc.) is the main reason for the large difference between

εsu,cy and εsu,nominal. By contrast, the prime reason for the - smaller, albeit significant - difference of

εsu,mon from εsu,nominal is statistical, similar to the size effect on strength. The 115 monotonic tests in

the database that failed by rupture of the tension steel had from 1 to 20 bars in the tension zone

and, unlike the cyclic tests, exhibit a statistically significant reduction of εsu at ultimate curvature

with increasing Nb,tension, as it is the minimum value of εsu among the bars that controls failure. The

functional form of Eq. (4.26b), namely the linearity in √lnN, is derived according to [4] as the

mode (i.e., most likely value) in a Type-I extreme value probability distribution of the smallest εsu

in N bars, all a mean εsu value of εsu,nominal. The parameters of the linear dependence are then fitted

to the test data. There are 5 tension bars on average in the 115 monotonic tests with rupture of

tension reinforcement and indeed Eq. (4.26b) degenerates to (4.26a) for Nb,tension=5. The statistics

at rows 9 and 10 in Table 4.1 show that Eq. (4.26b) gives lower prediction scatter than Eq. (4.26a).

30

Table 4.1: Statistics of test-to-predicted ultimate curvatures in uniaxial tests of members with continuous bars

φu,exp/φu,pred for different testing conditions and failure modes No of data


1 All uniaxial tests 474 1.105 0.995 49.7% 2 Monotonic tests 269 1.125 1.01 53.2% 3 Cyclic tests 205 1.08 0.985 44.2% 4 Tests without slippage of bars from the anchorage zone 349 1.135 0.995 50.4% 5 Tests with slippage of bars from the anchorage zone 125 1.03 0.98 46.4% 6 Failure at spalling of the full section 65 1.135 0.925 55.5% 7 Failure due to crushing of the confined core – monotonic tests 105 1.08 1.02 51.9% 8 Failure due to crushing of the confined core – cyclic tests 81 1.175 0.99 52.6% 9 Failure due to tension steel rupture – monotonic tests,Eq. (4.26a) 115 1.13 0.99 52.6%

10 Failure due to tension steel rupture– monotonic tests,Eq. (4.26b) 115 1.09 1.00 44.8% 11 Failure due to rupture of tension steel – cyclic tests 108 1.04 1.01 34.2%

* See footnote of Table 2.1.

4.3 Fixed-end rotation due to pull-out of longitudinal bars from their anchorage beyond

the end section at member flexural failure

After yielding of the end section and until ultimate curvature takes place there, inelastic strains

penetrate into the anchorage zone of tension bars beyond the end section, increasing the “fixed-

end rotation” due to their slippage from that zone. These bars are considered fully anchored

beyond the “yield-penetration-length”. In 125 test results among those used here for the

derivation/calibration of the models for ultimate curvatures, relative rotations measured at

different gauge lengths include the fixed-end rotation due to slippage of bars from their anchorage

zone. On this basis, at the same time the material ultimate strains of Eqs. (4.23)-(4.26) were

derived, the additional “fixed-end rotation” between yielding and ultimate curvature was inferred

as:

– cyclic loading: ubLslipu d ϕθ 5.5, =Δ (4.27a)

– monotonic loading: ubLslipu d ϕθ 5.9, =Δ (4.27b)

Eqs. (4.27) imply that the bars are perfectly-plastic along the “yield-penetration-length”. A slightly

better fit to the data is achieved if they are considered as linearly strain-hardening all along that

length. In that case ϕu is replaced in Eqs. (4.27) by (ϕy+ϕu)/2 but the “yield-penetration-length”

31

increases to 10dybL or 16dybL for cyclic or monotonic loading, respectively.

The fixed-end rotation from Eqs. (4.27) has been removed from these 121 ultimate curvature

measurements before comparing with predictions in Fig. 4.2 or deriving the statistics in Table 4.1.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

φu,

exp

(1/m

)

φu,pred (1/m)

slipno-slip median: φu,exp=φu,pred

Fig. 4.2 Experimental ultimate curvatures of members with continuous bars vs. prediction from

Eqs. (4.1)-(4.27) according to Flow Charts 1 and 2

4.4 Uniaxial ultimate chord rotation from curvatures and the plastic hinge length

The overall deformation measure used here for a member is the chord rotation at each end, θ, i.e.

the angle between the normal to the end section and the chord connecting the member ends at the

member’s displaced position. For flexure-controlled failure the plastic component of chord

rotation over the shear span Ls is often taken equal to the plastic component of the ultimate

curvature, ϕu-ϕy, times a plastic-hinge length, Lpl. Adding the fixed-end rotation due to bar

slippage from its anchorage beyond the member end, the ultimate chord rotation, θu, is:

slipusls

plplyuyu a

LL

L ,21)( θϕϕθθ Δ+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−+= (4.28)

with asl=0 if slippage of longitudinal bars from the anchorage is not physically possible and asl=1

if it is. Expressions for Lpl depend on the models used for θy, Δθu,slip, ϕu and ϕy. Considering the

comparisons so far with experimental data as confirmation of the models proposed here for Δθu,slip

32

and ϕu and for θy and ϕy, the empirical expressions in Eqs. (4.29) were derived for Lpl by fitting

Eq. (4.28) to the uniaxial test data to flexure-controlled ultimate chord rotation, using there the

values of ϕy and θy from Eqs. (2.1)-(2.6) and (2.13), respectively, of Δθu,slip from Eq. (4.27) and of

ϕu from Eqs. (4.1)-(4.26) and Flow Charts 1 and 2:

– For cyclic loading and proper detailing for earthquake resistance:

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛+=hL

hL scypl ;9min

3112.0, (4.29a)

– For monotonic loading, regardless of detailing: ⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛+=

hL

hL smonpl ;9min04.01.1, (4.29b)

Rows 1-7 in Table 4.2 give statistics of the ratio of experimental ultimate chord rotations to

the so-predicted values and Fig. 4.3 compares experimental to predicted values. The variance of

the data about Eq. (4.28) is close to 80% of the total. So, a better alternative is pursued below.

0

2.5

5

7.5

10

12.5

0 2.5 5 7.5 10 12.5

θ u,e

xp (%

)

θu,pred (%)

Cyclic loading

beams & columnsrect. wallsnon-rect. sections

median: θu,exp=θu,pred

5% fractile: θu,exp=0.45θu,pred

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35 40

θ u,e

xp (%

)

θu,pred (%)

Monotonic loading




(a) (b)

Fig. 4.3 Experimental ultimate chord rotations of members with continuous bars vs. predictions of

Eq. (4.28), using: (a) Eq. (4.29a) for cyclic loading; (b) Eq. (4.29b) for monotonic loading

4.5 Empirical uniaxial ultimate chord rotation

Three alternative empirical models, Eqs. (4.30), have been fitted to the ultimate chord rotation, θu,

of members with proper detailing for earthquake resistance carried to flexure-controlled failure by

33

cyclic or monotonic loading. Eqs. (4.30a), (4.30b) are improvements of empirical models

proposed in [19] on the basis of a smaller database. Eqs. (4.30b), (4.30c) give the plastic

component of θu, θupl=θu-θy, with the elastic component, θy, obtained from Eqs. (2.13):

( ) ( ) ( )( )

dc

yww

ff

scnrwrw

slcystu h

Lfaaaa ρ

αρ

ν

ωωαθ 100

35.0225.0

1

2,, 25.125;9min

;01.0max;01.0max3.0

72142.01

21)43.01(

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ +−=

(4.30a)

( ) dc

yww

f

f

sc

nrwrw

slcy

plst

plu h

Lfa

aaa ρ

αρ

ν

ωωαθ 100

35.02.0

3.0

1

2,, 275.125;9min

);01.0max();01.0max(25.0

41)44.01(

6.11)52.01(

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟

⎠⎞

⎜⎝⎛ +−=

(4.30b)

( ) ( ) dc

yww

f

f

cs

wslcy

hbwst

plu f

hL

bhaa ρ

αρ

ν

ωωαθ 1002.03

1

1

2 225.125;9min);01.0max();01.0max(2.0;10min;5.1max052.016.01)525.01(

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+−=

(4.30c) where:

– ast, aplst, ahbw

st: coefficients for the type of steel: for ductile hot-rolled or heat-treated steel

ast=aplst=0.0185 and ahbw

st=0.022; for cold-worked steel ast=0.0115, aplst=0.009, ahbw

st=0.0095;

– acy: zero-one variable for the type of loading, acy=0 for monotonic loading acy=1 for cyclic;

– asl: zero-one variable for slip, defined already for Eq. (4.28);

– aw,r: zero-one variable for rectangular walls, aw,r=1 for rectangular walls, aw,,r=0 otherwise;

– aw,nr: zero-one variable for non-rectangular sections, aw,nr=1 for T-, H-, U- or hollow

rectangular section and aw,nr=0 for rectangular ones;

– ν=N/bhfc with b: width of compression zone and N: axial force, positive for compression;

– ω1=(ρ1fy1+ρvfyv)/fc: total mechanical reinforcement ratio of tension and web longitudinal bars;

– ω2=ρ2fy2/fc: mechanical reinforcement ratio of the compression bars;

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

– α: confinement effectiveness factor from Eq. (4.21)

– ρw=Ash/bwsh: ratio of transverse steel parallel to the direction of the applied shear;

– fyw: yield stress of transverse steel;

– ρd: steel ratio of diagonal bars (if any) in each diagonal direction;

– bw: width of one web, even in cross-sections with one or more parallel webs.

34

Table 4.2: Statistics of test-to-predicted flexure-controlled ultimate chord rotations

Test-to-predicted ratio No of data

mean median C.o.V

1 θu,exp/θu,Eqs. (4.28), (4.29) – all uniaxial tests with continuous bars 1395 1.075 1.00 50.7%2 θu,exp/θu, Eqs. (4.28), (4.29b) – monotonic uniaxial tests with continuous bars 299 1.18 1.005 66.8%3 θu,exp/θu,Eqs. (4.28), (4.29a) – cyclic uniaxial tests for good detailing and continuous bars 1095 1.05 0.995 43.4%4 θu,exp/θu,Eqs. (4.28), (4.29) – uniaxial tests with continuous bars, without bar slip 213 1.145 1.015 64.1%5 θu,exp/θu,Eqs. (4.28), (4.29) – uniaxial tests with continuous bars, with bar slip 1181 1.055 0.99 46.8%6 θu,exp/θu,Eqs. (4.28),(4.29)–uniaxial tests - rectangular walls with continuous bars, with bar slip 88 1.24 1.08 51.1%7 θu,exp/θu,Eqs. (4.28), (4.29)–uniaxial tests– nonrectangular section, continuous bars and bar slip 56 1.095 1.065 45.9%

8 θu,exp/θu,Eq. (4.30a) – all uniaxial tests with continuous bars 1395 1.045 0.99 42.2%9 θu,exp/θu,Eq. (4.30a) – all monotonic uniaxial tests with continuous bars 299 1.14 1.00 53.3%

10 θu,exp/θu,Eq. (4.30a) – all cyclic uniaxial tests with good detailing and continuous bars 1095 1.025 0.99 37.3%11 θu,exp/θu,Eq. (4.30a) – all uniaxial tests w/ good detailing & continuous bars, without bar slip 213 1.10 0.98 50.1%12 θu,exp/θu,Eq. (4.30a) – all uniaxial tests w/ good detailing & continuous bars, with bar slip 1181 1.04 0.995 40.3%13 θu,exp/θu,Eq. (4.30a) – uniaxial tests - rectangular walls with continuous bars, bar slip 88 0.97 0.99 35.9%14 θu,exp/θu,Eq. (4.30a) – uniaxial tests - non-rectang. sections & continuous bars, bar slip 56 0.98 0995 32.8%

15 θu,exp/θu,Eq. (4.30b) – all uniaxial tests with continuous bars 1395 1.04 0.99 42.1%16 θu,exp/θu,Eq. (4.30b) – all monotonic uniaxial tests with continuous bars 299 1.13 0.995 53.5%17 θu,exp/θu,Eq. (4.30b) – all cyclic uniaxial tests with good detailing and continuous bars 1095 1.02 0.99 37.2%18 θu,exp/θu,Eq. (4.30b) – all uniaxial tests w/ good detailing & continuous bars, without bar slip 213 1.115 0.99 50.3%19 θu,exp/θu,Eq. (4.30b) – all uniaxial tests w/ good detailing & continuous bars, with bar slip 1181 1.03 0.99 40.0%20 θu,exp/θu,Eq. (4.30b) – uniaxial tests - rectangular walls with continuous bars, bar slip 88 0.965 0.96 36.1%21 θu,exp/θu,Eq. (4.30b) – uniaxial tests - non-rectang. sections & continuous bars, bar slip 56 0.965 0995 31.4%

22 θu,exp/θu,Eq. (4.30c) – all uniaxial tests with continuous bars 1395 1.055 0.995 42.5%23 θu,exp/θu,Eq. (4.30c) – all monotonic uniaxial tests with continuous bars 299 1.15 1.01 53.0%24 θu,exp/θu,Eq. (4.30c) – all cyclic uniaxial tests with good detailing and continuous bars 1095 1.03 0.99 38.0%25 θu,exp/θu,Eq. (4.30c) – all uniaxial tests w/ good detailing & continuous bars, without bar slip 213 1.14 0.99 50.2%26 θu,exp/θu,Eq. (4.30c) – all uniaxial tests w/ good detailing & continuous bars, with bar slip 1181 1.04 0.995 40.5%27 θu,exp/θu,Eq. (4.30c) – uniaxial tests - rectangular walls with continuous bars, bar slip 88 0.955 0.96 39.9%28 θu,exp/θu,Eq. (4.30c) – uniaxial tests - non-rectang. sections & continuous bars, bar slip 56 1.095 1.07 28.4%

29 θu,exp/θu,Eq. (4.31a) – cyclic uniaxial tests, continuous ribbed bars and poor detailing 48 1.04 1.005 30.6%30 θu,exp/θu,Eq. (4.31b),(4.30b) – cyclic uniaxial tests, continuous ribbed bars and poor detailing 48 1.00 0.99 31.8%31 θu,exp/θu,Eq. (4.31b),(4.30c) – cyclic uniaxial tests, continuous ribbed bars and poor detailing 48 1.01 1.01 30.2%32 θu,exp/θu,Eq. (4.32),(4.31a) – cyclic uniaxial tests, continuous smooth bars and poor detailing 32 1.035 1.015 34.5%33 θu,exp/θu,Eq. (4.32),(4.31b),(4.30b)–cyclic uniaxial tests, continuous smooth bars, poor detailing 32 1.01 1.01 32.2%34 θu,exp/θu,Eq. (4.32),(4.31b),(4.30c) – cyclic uniaxial tests, continuous smooth bars, poor detailing 32 1.005 1.015 31.7%

35 SRSS of θuy,exp/θuy,Eqs. (4.28), (4.29) & θuz,exp/θuz,Eqs. (4.28), (4.29) – continuous bars, biaxial tests 36 1.10 1.05 28.7%36 SRSS of θuy,exp/θuy,Eq. (4.30a) and θuz,exp/θuz,Eq. (4.30a) – continuous bars, biaxial tests 36 1.27 1.18 23.2%37 SRSS of θuy,exp/θuy,Eq. (4.30b) and θuz,exp/θuz,Eq. (4.30b) – continuous bars, biaxial tests 36 1.25 1.15 23.3%38 SRSS of θuy,exp/θuy,Eq. (4.30c) and θuz,exp/θuz,Eq. (4.30c) – continuous bars, biaxial tests 36 1.26 1.16 24.1%

39 θu,exp/θu,pred uniaxial with curvatures, pl. hinge length & Eq. (5.3) – lapped ribbed bars 81 0.98 0.98 36.6%40 θu,exp/θu,pred empirical Eq. (4.30b) and Eq. (5.4) – uniaxial tests, lap-spliced ribbed bars 81 1.06 1.035 39.3%41 θu,exp/θu,pred empirical Eq. (4.30c) and Eq. (5.4) – uniaxial tests, lap-spliced ribbed bars 81 1.045 1.045 39.0%42 θu,exp/θu,pred empirical Eq. (5.5) – uniaxial tests, lap-spliced smooth bars 11 1.19 1.03 33.4%

35

(a)

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35

θ u,e

xp (%

)

θu,pred (%)

Monotonic loading




0

2.5

5

7.5

10

12.5

0 2.5 5 7.5 10 12.5

θ u,e

xp (%

)

θu,pred (%)

Cyclic loading




(b)

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35

θ u,e

xp (%

)

θu,pred (%)

Monotonic loading




0

2.5

5

7.5

10

12.5

0 2.5 5 7.5 10 12.5

θ u,e

xp (%

)

θu,pred (%)

Cyclic loading

beams & columns

rect. walls

non-rect. sections



(c)

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35

θ u,e

xp (%

)

θu,pred (%)

Monotonic loading




0

2.5

5

7.5

10

12.5

0 2.5 5 7.5 10 12.5

θ u,e

xp (%

)

θu,pred (%)

Cyclic loading

beams & columns

rect. walls

non-rect. sections



Fig. 4.4 Experimental ultimate chord rotation of members with continuous bars vs. predictions of:

36

(a) Eq. (4.30a); (b) Eq. (4.30b); (c) Eq. (4.30c)

Eqs. (4.30b) or (4.30c) can more readily be extended for lap-splicing of bars in the plastic hinge

region (chapter 5) and/or wrapping of the end(s) with FRP (chapter 8), which affect differently the

values of θupl and θy. Eq. (4.30c) distinguishes walls or members with T-, H-, U- or hollow

rectangular section via the slenderness ratio, h/bw, of the web. Fig. 4.4 and the statistics in rows 8-

28 of Table 4.2 show that the three versions of Eqs. (4.30) provide practically the same accuracy

and much smaller scatter than Eqs. (4.28), (4.29). The variance of the monotonic and the cyclic

data with respect to Eqs. (4.30) is about 40% or 45% of their corresponding total variance.

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9

θ u,e

xp (%

)

θu,pred (%)


(a)

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9

θ u,e

xp (%

)

θu,pred (%)

median: θu,exp=0.99θu,pred

(b)

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9

θ u,e

xp (%

)

θu,pred (%)


(c)

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

θ u,e

xp (%

)

θu,pred (%)


(d)

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

θ u,e

xp (%

)

θu,pred (%)


(e)

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

θ u,e

xp (

%)

θu,pred (%)


(f) Fig. 4.5 Experimental cyclic ultimate chord rotations of members with poor detailing and (a)-(c)

ribbed or (d)-(f) smooth continuous bars bars, vs. predictions of: (a) Eq. (4.31a); (b) Eqs. (4.31b),

(4.30b); (c) Eqs. (4.31b), (4.30c); (d) Eqs. (4.32), (4.31a); (e) Eqs. (4.32), (4.31b) and (4.30b); (f)

Eqs. (4.32), (4.31b) and (4.30c)

About 50 tests of members with continuous ribbed longitudinal bars and about 30 tests with

37

smooth ones, all without seismic detailing (i.e., with sparse ties without 135o hooks) show that

poor detailing does not affect the flexure-controlled ultimate chord rotation for monotonic loading,

but it does for cyclic as follows:

– Old-type members with ribbed bars, cyclic loading: θu = θu,Eq. (4.30a)/1.2, or (4.31a)

θupl

= θupl

Eq. (4.30b) or (4.30c)/1.2 (4.31b)

– Old-type members with smooth bars, cyclic loading: θu = 0.95θu,Eq. (4.31) (4.32)

Fig. 4.5 compares test results to the predictions of Eqs. (4.31), (4.32). The statistics of the test-

to-prediction ratio are given in rows 29-34 of Table 4.2.

Eqs. (4.30a), (4.30b) and (4.31) - with minor variations in some coefficients - have been

adopted in Part 3 of Eurocode 8 [11] for beams and rectangular columns or walls under cyclic

loading.

4.6 Flexure-controlled ultimate chord rotations in biaxial loading

The few available biaxial tests suggest that at ultimate deformation the chord rotation components

parallel to the sides of the cross-section, θuy and θuz, lie on average from 5% to 18% outside a

circular interaction diagram:

12

,

2

,=⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

uniuz

uz

uniuy

uy

θθ

θθ

(4.33)

θuy,uni and θuz,uni are the ultimate chord rotations under unidirectional bending parallel to the

section sides computed from Eqs. (4.28), (4.29) or via Eqs. (4.30). The statistics at rows 35-38 in

Table 4.2 suggest that the alternative uniaxial models are effectively equivalent for use in Eq.

(4.33). The smaller scatter compared to the uniaxial data is due to the fewer biaxial ones.

38

5 Effect of lap-splicing of longitudinal bars in the plastic hinge zone on

flexure-controlled ultimate deformations

Fig. 5.1 shows the ratio of the experimental plastic part of ultimate chord rotation, θplu, to the value

from Eq. (4.30b), while Fig. 5.2 depicts the strain of the lapped tension bars at which the

experimental ultimate chord rotation is attained, normalized to their ultimate strain, εsu. It is shown

directly in Fig. 5.1 and indirectly in Fig. 5.2 that, if the lap length is relatively long, both

approaches underestimate the experimental ultimate chord rotation of members with lap-spliced

bars. To reflect this finding and account - at least partly - for this difference, it is proposed to

consider as compression reinforcement for ultimate curvatures or chord rotation models both bars

in any pair of lapped bars in the compression zone.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4

θ pl,e

xp /θ

pl,th

lo/lou,min

beams & columns


Fig. 5.1. Ratio of experimental plastic part of ultimate chord rotation, θplu, of members with lap-

spliced bars to empirical prediction neglecting the lap splicing, compared to Eq. (5.4)

The scatter notwithstanding, Figs. 5.1 and 5.2 suggest that if the lap length, lo, is less than a

minimum lap length, lou,min, beyond which the ultimate chord rotation does not seem to be

adversely affected by the lap splice, both θplu and the maximum usable tensile strain decrease with

decreasing lo. This minimum lapping is longer than that required for the member to develop its full

yield moment, loy,min, given by Eq. (3.4). It has also been found to be positively affected by

39

clamping of the lapped bars by transverse reinforcement, as:

cc

ywwsl

ybLou

fff

a

fdl

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=ρ

,

min,

5.1405.1

(fy, fyw, fc in MPa) (5.1)

where:

− ρw: ratio of transverse reinforcement parallel to the plane of bending, and

− al,s=(1-0.5sh/bo)(1-0.5sh/ho)nrestr/ntot (5.2)

with sh, bo, ho as for Eq. (4.21) and:

• ntot: total number of lapped bars in the cross-section,

• nrestr: number of these bars which are engaged by a stirrup corner or a cross-tie.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ε su,

"exp

" /ε

su,th

lo/lou,min

beams&columnsT,H,U or hollow rect.

Fig. 5.2. Ratio of strain in tension bars at which the experimental ultimate chord rotation of a

member with lap-spliced bars is attained to the ultimate bar strain, compared to Eq. (5.3)

The elastic component, θy, of the ultimate chord rotation is modified according to Section 3.2

above, to account for the lap splicing. On the basis of Figs. 5.1 and 5.2, it is proposed here to

modify the plastic component of θu as follows:

1. If Eqs. (4.28), (4.29) are used, only the calculation of curvatures is modified for the effect of

lap splicing. The yield curvature, φy, is modified according to section 3.2 above. For φu, both

lapped compression bars in any pair are included in the compression reinforcement. Moreover,

40

if lo is shorter than the value of lou,min from Eq. (5.1), the maximum elongation of the extreme

tension bars at ultimate conditions due to steel failure is reduced to the following limit value:

s

y

ou

osu

ou

olsu E

fl

ll

l

min,min,, 2.02.1 ≥⎟

⎟⎠

⎞⎜⎜⎝

⎛−= εε (5.3)

with εsu from Eq. (4.25) for cyclic loading or (4.26) for monotonic and loy,min, lou,min given by

Eqs. (4.20) and (5.1) above. Everything else in this approach, namely Eqs. (4.1)-(4.29), still

apply.

2. If Eqs. (4.30b), (4.30c) are adopted, the plastic part of the ultimate chord rotation is estimated

as:

plu

ou

oplbarslappedribbedu l

lθθ ⎟⎟⎠

⎞⎜⎜⎝

⎛=

min,, ,1min (5.4)

with θplu calculated from Eqs. (4.30b) or (4.30c) as if the bars were continuous.

0

2

4

6

8

10

0 2 4 6 8 10

θ u,e

xp(%

)

θu,pred (%)

beams & columnsT,H,U or hollow rect.


(a)

0

2

4

6

8

10

0 2 4 6 8 10

θ u,e

xp(%

)

θu,pred (%)

beams & columnsT,H,U or hollow rect.


(b) Fig. 5.3. Experimental ultimate chord rotations of members with lap-spliced bars vs. predictions

of: (a) the empirical approach and Eq. (5.4); (b) the approach based on curvatures, the plastic

hinge length and Eq. (5.3)

Fig. 5.3 compares the data to the outcome of the option 1 (in Fig. 5.3(b)) or of option 2 (in Fig.

5.3(a)). Rows 39 to 41 in Table 4.2 give statistics of the corresponding test-to-prediction ratios.

Note that, although for members with continuous bars Eqs. (4.30) give lower prediction scatter

41

than Eqs. (4.28), (4.29), their extensions for lap-spliced bars are practically equivalent.

Eqs. (5.2) and (5.4) have been adopted in Part 3 of Eurocode 8 [11].

There are just 7 cyclic tests of columns without detailing for earthquake resistance and smooth

hooked bars lapped starting at the base (all with lo≥15dbL). They suggest the following

modification of Eq. (4.32) for the chord rotation at flexure-controlled failure:

, , .(4.32)

10 min ; 40

50

o

bLu smooth lapped bars u Eq

ld

θ θ

⎛ ⎞+ ⎜ ⎟

⎝ ⎠= (5.5)

The last row in Table 4.2 gives statistics of the corresponding test-to-prediction ratios.

42

6 Shear Strength after Flexural Yielding

6.1 Introduction

If it takes place before flexural yielding, ultimate failure of concrete members in shear occurs at

relatively low deformations and is considered as a “brittle” failure mode. Sometimes concrete

members that yield first in flexure, may ultimately fail under cyclic loading with their failure

mode showing strong and clear effects of shear. Notably inclined cracks increase in width and

extent with cycling despite the gradual drop of peak force resistance with load cycling; on the

other hand, phenomena which are normally associated with flexural failure, such as a single wide

crack transverse to the axis at the section of maximum moment, disintegration of the compression

zone and/or buckling of longitudinal bars next to the section of maximum moment, are not

pronounced. Close to a flexure-controlled ultimate condition, these latter phenomena are

dominant, sometimes leading to rupture of a longitudinal bar, whereas the width of any inclined

cracks that may have developed at the beginning decreases and such cracks may even disappear,

as the peak force resistance drops with load cycling after the flexure-controlled ultimate strength.

Failure in shear under cyclic loading, after initial flexural yielding is termed “ductile shear” failure

[14]. It is normally associated with 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 [3], [14], [17].

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

concrete 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.

43

6.2 Models of shear resistance in diagonal tension under inelastic cyclic deformations after

flexural yielding

An earlier work [7] used a fairly large data set of columns with circular or rectangular section, and

of walls, ultimately failing by “ductile shear” under cyclic loading, to develop two models for the

shear resistance, VR, as a function of the plastic chord rotation (or displacement) ductility ratio,

μplθ, 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. (2.13). In both models the effect of axial

force, N, on VR is accounted for through a separate term. That term represents the contribution to

shear resistance of the transverse to the member axis component of the compression strut between

the two ends of the member (as in [12], as well as in [14]). A 45o truss inclination is considered, as

in [17], 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).

( ) wccs

totplθcc

sR VAf

hLρμfAN

LxhV +⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞⎜

⎝⎛ ⎟

⎠⎞⎜

⎝⎛−⋅+

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

2 (6.1a)

( ) ( )( ) ⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−−+−

= wccs

totpl

ccs

R VAfhL

fANL

xhV ,5min16.01)100,5.0max(16.0,5min05.0155.0,min2

ρμθ (6.1b)

where:

h: depth of cross-section;

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, equal to bwd for cross-sections with rectangular web of width (thickness)

bw and structural depth d;

fc: concrete strength (ΜPa);

ρtot: total longitudinal reinforcement ratio;

Vw: contribution of transverse reinforcement to shear resistance, taken equal to:

44

ywwww zfbV ρ= (6.2)

where:

bw is the width (thickness) of the rectangular web,

ρw and fyw are the ratio and yield stress of transverse reinforcement, and

z is the internal lever arm (z = d-d1 in beams, columns and walls of barbelled, T- or H-shaped

section, z = 0.8h in rectangular walls),

In [7] Eqs. (6.1) were fitted to a database of 239 cyclic tests that led to “ductile shear” failure

after initial flexural yielding; this total included 53 tests on columns with circular section, 161

tests on columns or beams with square or rectangular section, 6 on walls and 19 on piers with

hollow or T-shaped section. As far as the fitting is concerned, Eqs. (6.1a) and (6.1b) are practically

equivalent: the ratio of experimental-to-predicted shear resistance had a median of 1.0 for both

models and a coefficient of variation of 15.1% for Eq. (6.1a) or 14.1% for (6.1b); Eq. (6.1b) also

had slightly better average agreement to the data for each one of the four types of members

(circular or rectangular columns, rectangular walls and non-rectangular walls or piers).

Since the earlier work the database of cyclic “ductile shear” failures after initial flexural

yielding has been enriched with 18 more tests on columns with circular section, 32 on rectangular

columns, 5 on rectangular walls and 12 more tests on non-rectangular walls. In the present report

Eqs. (6.1) are evaluated on the basis of the increased databank of beams and columns with

rectangular section and piers with hollow rectangular, T, H, or U section. More specifically, for

this type of members, 235 in total, statistics of the ratio of experimental-to-predicted shear

resistance are computed for the increased database, as follows:

For Eq. (6.1a): median and coefficient of variation of the ratio of experimental-to-predicted

shear resistance in the 235 tests equal to 0.985 and 15.05% respectively;

For Eq. (6.1b): median and coefficient of variation of the ratio of experimental-to-predicted

shear resistance in the 235 tests equal to 0.99 and 14.25% respectively.

45

Fig. 6.1 compares the predictions of Eqs. (6.1a), (6.1b) to the experimental values of shear

resistance in the 235 tests.

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

V exp

(kN

)

Vpred (kN)

beams & columns

rectangular walls

piers with T,H,U or hollow rect. section

(a)

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000V e

xp(k

N)

Vpred (kN)

beams & columns

rectangular walls

piers with T,H,U or hollow rect. section

(b) Fig. 6.1. Experimental vs. predicted shear strength for tests failed in shear due to diagonal tension;

(a) Eq. (6.1a); (b) Eq. (6.1b).

46

7 RC-jacketing of columns

7.1 Introduction

Jacketing a column is the most cost-effective way to enhance at the same time:

1. its flexural resistance (even converting a weak-column/strong-beam frame into a strong-

column/weak-beam one),

2. its lateral stiffness,

3. its shear strength,

4. its deformation capacity and

5. anchorage and continuity of reinforcement in anchorage or splicing zones.

Being a composite member of two different concretes and two distinct cages of reinforcement

with different detailing and often different types of steel, a jacketed member seems fairly complex

and difficult to tackle in everyday retrofit design. The uncertain behaviour of the interface between

the old member and the jacket adds to this difficulty. To reduce this problem to a level of

simplicity consistent with the popularity of concrete jackets as an easy and low cost retrofitting

technique, simple rules are developed for the estimation of the yield moment, the yield drift, the

secant stiffness at incipient yielding and the flexure-controlled cyclic deformation capacity of

jacketed members. To this end data from 57 monotonic or cyclic tests (55 on columns and two on

walls) of members jacketed with concrete employing various bonding measures at the interface

between the old and the new concrete, have been used to express the properties of the jacketed

column in terms of the corresponding property of an “equivalent” monolithic member.

7.2 Simple rules for the strength, the stiffness and the deformation capacity of jacketed

members

The rules proposed here on the basis of the tests in the database use modification factors on the

properties of an “equivalent” monolithic member. The strength, the stiffness and the deformation

capacity of the “equivalent” monolithic member are determined according to the rules above and

47

to the additional considerations listed in Table 7.1. The idea behind assumptions A.3 and A.4 in

this Table is that, for common ratios of jacket thickness to depth of the jacketed section, it is

mainly the jacket that carries the full axial load at the critical end section and in the plastic hinge

of the column. Also, it is the jacket that mainly controls the shear resistance and the bond along

the longitudinal reinforcement of the jacket.

In the following, an asterisk is used to denote a calculated value for the jacketed member, as,

e.g., in My*, θy

*, θu*. No asterisk is used (as, e.g., in My, θy, θu

pl) for values calculated for the

monolithic member according to the assumptions in Table 7.1 and chapters 2 and 4. Ratios of

experimental values of My, θy and θu for the 57 jacketed members in the database to the values of

My, θy and θu calculated for the monolithic member according to the assumptions in Table 7.1 and

chapters 2 and 4 are shown in Fig. 7.1. Note that in Fig. 7.1 (bottom) θu,cal is taken equal to

θy*+θu

plEq. (4.30), with θy

* = 1.05θy,Eq. (2.13) being the overall best estimate of the chord rotation at

yielding for the jacketed member (with θy from Eqs. (2.13)). With so defined θy*, in Fig. 7.1 (3rd

figure from the top) the effective stiffness to yield point of the jacketed member is defined as:

EI*eff = My,calLs/3θy

*. The ratios My,exp/My,pred, θy,exp/θy,Eq. (2.13), EIexp/EI*eff and θu,exp/θu,cal are given

from top to bottom of Fig. 7.1 separately for different ways of connecting the jacket to the old

member and separately for those members which had been damaged by testing before they were

jacketed. Specimens in which the longitudinal reinforcement of the jacket did not continue beyond

the member end, or specimens with lap-spliced reinforcement in the original member, are

identified in Fig. 7.1 but otherwise lumped together with those tests where the vertical bars in the

original member were continuous. For tests that did not reach ultimate conditions and for the two

walls that failed in the unstrengthened part of their height, an arrow pointing up signifies an

experimental-to-predicted ratio greater than the plotted value.

48

Table 7.1 Assumptions for the properties of a monolithic member considered as "equivalent" to the jacketed one

I. Flexural resistance and deformation capacity, deformations at flexural yielding Case A: Jacket longitudinal bars are anchored beyond the member end sections:

A1: Dimensions External dimensions of the section are those of the jacket. A2: Longitudinal reinforcement

The tension and compression reinforcement are those of the jacket. Longitudinal bars of the old member are considered at their actual location between the tension and compression bars of the jacket: − they may supplement longitudinal bars of the jacket between the

tension and compression reinforcement and be included in a uniform “web” reinforcement ratio between the tension and compression bars

− in walls the tension and compression reinforcement of the jacketed member may include old vertical bars at the edges, as appropriate.

Lap splices in the intermediate old reinforcement may be neglected. Differences in yield stress between the new and old longitudinal reinforcement should be taken into account, in all cases.

A3: Concrete strength The fc value of the jacket applies over the full section of the monolithic member; in the 3rd term of Eqs. (2.13) the fc value of the concrete into which the longitudinal bars are anchored beyond the end section is used.

A4: Axial load The full axial load is taken to act on the jacketed column as a whole, although it was originally applied to the old column alone.

A5: Transverse reinforcement

Only the transverse reinforcement in the jacket is taken into account for confinement.

Case B. Jacket longitudinal reinforcement stops at the end section: B1: Dimensions, longitudinal reinforcement, concrete strength

My and ϕy (also in the 1st and 3rd term of Eqs. (2.13)) are calculated using the cross-sectional dimensions, the longitudinal reinforcement and the fc value of the old member, neglecting any contribution from the jacket. The effect of lap splicing of the old bars is taken into account as in chapters 3 and 5 for non-retrofitted members. The section depth h in the 2nd term of Eqs. (2.13) is that of the jacket.

B2: Transverse reinforcement

The deformation capacity, θu, is calculated on the basis of the old column alone, with the old column taken as confined by the jacket and its transverse steel. The value of ρs = As/bws for Eqs. (4.30) is determined using the value of As/s in the jacket and taking as bw the width of the old column. The confinement effectiveness factor may be taken αs = 1.0.

II. Shear resistance Shear resistance (including that without shear reinforcement, VR,c, for the determination of the value of αV in the 1st term of Eqs. (2.13)) and anything that has to do with shear is calculated on the basis of the external dimensions and the transverse reinforcement of the jacket. The contribution of the old transverse reinforcement may be considered only in walls, provided it is well anchored into the (new) boundary elements.

The average value and ± standard-deviation estimates of the mean test-to-prediction ratios are

shown in Fig. 7.1 separately for the various groups of specimens representing different types of

49

jacket-to-old-member connection, with or without damage in the original column. Note that, the

distance of the sample average from a certain reference value (e.g. 1.0), normalised by the

standard-deviation of the mean is a criterion on whether the value of the property of the jacketed

member may be taken equal to that calculated for the monolithic member according to the

assumptions in Table 7.1 and Sections 2 and 4, times that reference value.

Fig. 7.1 supports the following rules for calculating the yield moment, the chord rotation at the

yield point and the ultimate chord rotation, My*, θy

*, θu*, respectively, of the jacketed member, in

terms of the values My, θy, θupl calculated for the monolithic member according to Table 7.1 and

chapters 2 and 4:

1. For My*:

My* = My,pred (7.1)

2. For θy* (the main target being the stiffness at yield point, EI*

eff = My,calLs/3θy* with My

* =

My,pred), irrespective of any pre-damage in the original column:

θy* = 1.05θy,Eq. (2.13) (7.2)

(In [11] this rule has been adopted only for a roughened interface of the jacket to the old

concrete, with or without dowels, but the more conservative rule: θy* = 1.2θy,Eq. (2.13) has been

adopted [11] for no treatment of the interface, or dowels alone, or jacket bars connected to the

old ones via welded U-bars).

3. For θu*:

θu*= θ*

y+θupl

Eq. (4.30b) (7.3)

Rules 1 to 3, supplemented with assumptions B1 and B2 in Table 7.1, apply also if the jacket

longitudinal bars stop at the end section of the member.

If no differentiation is made for the measure taken to enhance the shear transfer at the interface

of the old and the new concrete, the ratio of the experimental value to the prediction from rules 1

to 3 above has overall median value and coefficient-of-variation equal to 1.035 and 10.7%, 0.99

50

and 23.5%, 1.005 and 30.5%, and 1.145 and 19% for My, θy, EIeff and θu, respectively.

0

0.2

0.4

0.6

0.8

1

1.2

M y,e

xp /

M y,c

alc

continuous bars smooth 15db laps ribbed 15db laps

smooth 25db laps ribbed 30db laps ribbed 45db laps

non-anchored jacket bars group average st. dev. of group mean

j

Legend: a: no treatment of interface b: no treatment, pre-damaged member, c: welded U-bars, d: dowels, e: roughened interface, f: roughened interface, member pre-damaged, g: U-bars and roughened interface, h: U-bars and roughened interface, member pre-damaged, i: dowels and roughened interface, j: dowels and roughened interface, member pre-damaged, k: monolithic member

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

θ y,e

xp / θ

y,cal

c

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

EI ex

p / E

I* eff

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

θ u,e

xp /

( θ*y

+ θ

upl )

a b c d e f g h i j k

Fig. 7.1 Experimental value for the RC-jacketed member divided to value calculated for the

monolithic member according to Table 7.1

51

Bonding measures at the interface of the jacket and the old member seem to have a statistically

significant effect only on the ultimate chord rotation, θu. The proposed rules underestimate on

average the measured ultimate chord rotation, θu,exp, for roughening and/or dowels at the interface

or for U-bars welded to the new and the old longitudinal bars. Even when no measure is taken to

improve the interface between the old and the new concrete or connect the two materials there, the

predictions undershoot the ultimate chord rotation of the jacketed member, but by less. So, it is

safe-sided for the ultimate chord rotation, θu, to neglect the favourable effect of positive

connection measures at the interface of the old and the new concrete, underestimating its measured

value by 14.5% on average. No systematic positive effect of any connection measures on the yield

moment, My, and the effective stiffness, EIeff, has been found.

The values of My, θy, EIeff and θu predicted for the 57 jacketed specimens in the database as

My*, θy

*, EI*eff or θu

* above according to rules 1 to 3 above do not show a systematic bias with

respect to any of the following:

− the ratio of fc of the jacket to that of the old member;

− the ratio of the cross-sectional area of the jacket to that of the old member;

− the ratio of the yield stress times the longitudinal reinforcement ratio in the jacket, to the same

product in the old member;

− the axial load, normalised to either the product of the full cross-sectional area of the jacketed

section and of fc of the jacket, or to the actual compressive strength of the jacketed section; and

− the ratio of the neutral axis depth at yielding to the thickness of the jacket.

The data do support assumptions A3 and A4 in Table 7.1, even when the compression zone

extends beyond the jacket, into the section of the old column.

The 57 jacketed specimens in the database did not show any shear distress at failure. This is

consistent with the fact that in all tests the shear resistance from Eqs. (6.1) was higher by at least

30% than the maximum applied shear force.

52

8 FRP-jacketing of columns

8.1 Seismic retrofitting with FRPs

Externally bonded Fibre Reinforced Polymers (FRPs) are used in seismic retrofitting in order to

enhance or improve:

i. the deformation capacity of flexural plastic hinges (with the fibres along the perimeter of the

section and FRP wrapping all-along the plastic hinge);

ii. deficient lap splices (with the fibres as in i and the FRP at least over the full lap length);

iii. the shear resistance (with the fibres in the transverse direction where enhancement of shear

strength is pursued).

FRPs do not lend themselves for enhancement of flexural resistance against seismic actions even

when their fibres are in the longitudinal direction of the member, as they cannot easily be

continued into the joint beyond the member end where the seismic bending moment is maximum.

8.2 FRP-wrapped columns with continuous vertical bars

8.2.1 Yield moment and effective stiffness to yield point

Eqs. (2.1) - (2.7) can be applied also to members with FRP-wrapping of their end regions, but with

the following modifications:

In the calculation of the values of φy and My of FRP-wrapped columns on the basis of 1st

principles, the unconfined concrete strength, fc, is replaced by the value fc* increased due to FRP

confinement according to [15],[16]:

( )( ) c

fuff

yx

yx

c

cff

abbbb

ff ,

2*

;max;min

3.31ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛+= (8.1)

where:

• bx and by are the cross-sectional dimensions of the section,

• ρf is the geometric ratio of the FRP parallel to the direction of bending,

• αf is the effectiveness factor for confinement by FRP of a rectangular section having its corners

53

rounded to a radius R to apply the FRP:

yx

yxf bb

RbRba

3)2()2(

122 −+−

−= (8.2)

• fu,f is the effective strength of the FRP, taken according to [15],[16] as equal to:

fu,f = Ef(keffεu,f) (8.3)

where:

− Ef and εu,f are the FRP’s Elastic modulus and failure strain, respectively,

− keff is an FRP effectiveness factor, taken according to [15],[16] as equal to keff = 0.6 for

Carbon FRP (CFRP) or Glass FRP (GFRP); for Aramid FRP (AFRP) and FRPs with

polyacetal fibres the value of keff is taken here the same as for CFRP and GFRP (keff = 0.85

has been proposed in [15],[16] for AFRP on the basis of few test results).

The increase of concrete strength according to Eq. (8.1) is not sufficient to capture the

enhancement of yield moment due to the confinement by FRP: as shown in Fig. 8.1(a) and at the

1st row of statistics in Table 8.1 (at the end of Section 8), the value of My computed on the basis of

first principles is, on average, 6.5% less than the experimental value. So, when Eqs. (2.13) is

applied to members with FRP-wrapped ends using a value of φy from 1st principles, a coefficient

of 1.065 should be applied on the 1st (flexural) term. The so-computed value of θy is compared in

Fig. 8.1(b) to test results for not-pre-damaged columns wrapped with FRP.

The effective stiffness from Eq. (2.14) using the value of My from 1st principles and that of θy

from Eq. (2.13), with the 1st term incorporating the factor 1.065 of the paragraph above, is

compared in Fig. 8.2 to experimental values. Table 8.1 gives also the statistics of the test-to-

prediction ratio for the effective stiffness at yielding for FRP-wrapped columns.

Fig. 8.1(b) and Table 8.1 (at the end of Section 8) show also the effect of serious previous

damage (from yielding to exceedance of ultimate deformation) before repair, FRP-wrapping and

re-testing. Such columns have also been included in the comparisons in Fig. 8.1(a) and in the 1st

row of statistics in Table 8.1, showing that repair of the damage and FRP-wrapping fully re-

54

instates the yield moment. However, Figs 8.1(b) and 8.2, as well as the 4th and 8th rows of statistics

in Table 8.1, suggest that, despite the repair and the FRP-wrapping, previous damage markedly

reduces the effective flexural stiffness to the yield point.

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500

My,

exp

(KN

m)

My,pred (KNm)


(a)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3

θ y,e

xp(%

)

θy,pred (%)

non-predamaged

predamaged

median (non-predamaged): θy,exp=0.965θy,pred

(b)

Fig. 8.1 FRP-wrapped rectangular columns with continuous bars: (a) experimental yield moment v

prediction from 1st principles and Eqs. (8.1), (8.2); (b) experimental chord rotation at yielding v

prediction of Eq. (2.13)

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350

EIex

p(M

Nm

2 )

EIpred (MNm2)

non-predamagedpredamaged

median (non-predamaged): EIexp=1.03EIpred

(a)

0

2

4

6

8

0 2 4 6 8

EIex

p(M

Nm

2 )

EIpred (MNm2)

non-predamagedpredamaged

median (non-predamaged): EIexp=1.03EIpred

(b)

Fig. 8.2 (a) Experimental stiffness to yield point of FRP-wrapped rectangular columns with

continuous bars, v effective stiffness from Eq. (2.14) with My from first principles and θy from Eq.

(2.13); (b) detail of (a).

55

8.2.2 Flexure-controlled deformation capacity

As in Section 4.5, the ultimate chord rotation, θu, can be expressed as the chord rotation at

yielding, θy, computed as in Section 8.2.1, plus a plastic part, θupl.

Two alternative types of model are proposed here for θupl. In the first one, θu

pl is taken equal to

the plastic component of the ultimate curvature, φu-φy, times a plastic-hinge length, Lpl, plus a

fixed-end rotation due to bar pull-out from the anchorage zone past the member end. A large

volume of data on flexure-controlled ultimate curvatures and chord rotations of rectangular

members without FRP-wrapping suggest a fixed-end rotation equal on average to a yield

penetration depth of 10 bar-diameters times the average of φy and φu (corresponding to linear

strain hardening of the bar along the yield penetration depth):

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−−+

+=

s

plplyubL

yusl

plu L

LLda

21)(10

2ϕϕ

ϕϕθ (8.4)

Empirical expressions for Lpl depend on the models used for φy and φu. The model used here for φy

is the one used for Eqs. (2.1)–(2.7), based on plane-section analysis and a strength of FRP-

confined concrete, fc*, from Eq. (8.1). The model for φu is also based on plane-section analysis as

explained in Section 4.1. For flexural failure in cyclic loading due to rupture of the extreme

tension bars it uses a limit strain, εsu, equal to 3/8 the steel uniform elongation at ultimate strength,

Eq. (4.25). The concrete σ-ε law is taken as parabolic-trapezoidal with ultimate strength from Eqs.

(8.1)-(8.3) [15],[16]. If the ultimate strain, εcu*, is also taken according to [15],[16], the flexure-

controlled ultimate curvature and chord rotation of rectangular FRP-wrapped members is

considerably under-estimated. So, a different ultimate strain value has been fitted to these data:

2,*

,*

100.0035 0.4 min 0.5;( )

f u fcu f eff j

c

fa a

h mm fρ

ε⎡ ⎤⎛ ⎞

= + +⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

(8.5)

Eq. (8.5) is a modification/extension of Eq. (4.23), fitted to a very large database of non-

wrapped members failing in flexure under cyclic loading. In Eq. (8.5) the section depth, h, is in

mm and ρf, af, fu,f were defined above via, or in conjunction with, Eqs. (8.1)-(8.3). The additional

56

parameter is another effectiveness factor for the FRP jacket, expressing that its effectiveness is not

proportional to the geometric ratio and stiffness of the FRP:

− ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−= *

,, ;5.0min15.0

c

fufjeff

f

fa

ρ for CFRP, GFRP, (8.6a)

− ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−= *

,, ;5.0min13.0

c

fufjeff

f

fa

ρ for AFRP (8.6b)

Note that, if the FRP provides relatively light confinement compared to the transverse

reinforcement, the end section may survive rupture of the FRP jacket and attain later a larger

ultimate curvature controlled by the confined concrete core inside the stirrups. In that case,

ultimate curvature can be calculated according to Section 4.1 for non-retrofitted members.

It has been proposed in [6], [8] and adopted in [11] to extend the empirical model for θupl, Eq.

(4.30b), to members with FRP wrapping by including in the exponent of the 2nd term from the end

the effect of confinement by the FRP, adding to it the term afρf ff,e, where ρf and af were defined

above via, or in conjunction with, Eqs. (8.1), (8.2), and ff,e, is the effective stress of the FRP:

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−=

c

fffunomfuffunomfuef f

EfEffρ

εε ,,,,, ;min7.0;5.0min1;min (8.7)

with ffu,nom denoting the nominal strength of the FRP and εu,f being a limit strain:

• εu,f = 0.015 for CFRP or AFRP;

• εu,f = 0.02 for GFRP.

It is proposed here to improve the extension of Eq. (4.30b) by adding to the exponent of the 2nd

term from the end of Eq. (4.30b) a term for the FRP symbolized by the left-hand-side of the

following expression and given by its right-hand-side:

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡=⎟⎟

⎠

⎞⎜⎜⎝

⎛

c

fffunomfu

c

fffunomfuf

efffc

u

fEf

fEfa

ffa ρ

ερ

ερ

,,,,,

;min;0.1min4.01;min;0.1min (8.8)

with the limit strain always equal to εu,f = 0.015. About the same fit to the tests is achieved if the

FRP-confinement term added to the exponent of the 2nd term from the end of Eq. (4.30b) is based

57

on the effective FRP strength in Eq. (8.3). This alternative, which is more consistent with the

confinement model in [15],[16], is:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡=⎟⎟

⎠

⎞⎜⎜⎝

⎛

c

fuf

c

fufff

efffc

u

ff

ff

caffa ,,

,

;4.0min5.01;4.0minρρρ (8.9)

where cf = 1.8 for CFRP and cf = 0.8 for GFRP or AFRP.

The last term in each one of Eqs. (8.6), Eq. (8.7)-(8.9) reflects the experimentally documented

reduced effectiveness of larger amounts of FRP wrapping.

Figs. 8.3, 8.4 compare the predictions of the models for θu (Eqs. (4.28), (8.1)-(8.6), or Eqs.

(2.13), (4.30b) with Eqs. (8.7), (8.8), or (8.9)) to the test results for FRP-wrapped columns to

which the models were fitted. Rows 10 to 13 in Table 8.1 (at the end of Section 8) refer to the test-

to-prediction ratios of θu for specimens without pre-damage, rows 14 to 17 for pre-damaged ones

and rows 18 to 21 to all specimens, regardless of pre-damage. The results of Eqs. (4.28), (8.1)-

(8.6) show no evidence of an effect of pre-damage on ultimate chord rotation. Eqs. (8.7), (8.8), or

(8.9), by contrast, do suggest a reduction of θu of about 10% due to pre-damage.

0

5

10

15

20

25

0 5 10 15 20 25θu,pred (%)

θ u,e

xp (%

)

CFRP jacketAFRP jacketGFRP jacketPAF jacket


(a)

0

5

10

15

20

25

0 5 10 15 20 25θu,pred (%)

θ u,e

xp (%

)



(b)

Fig. 8.3 Experimental ultimate chord rotation of FRP-wrapped rectangular columns with

continuous bars v predictions: (a) of model based on plastic hinge length, Eqs. (8.1)-(8.6) and

(4.28); (b) of empirical model, Eqs. (4.30b), (8.7).

58

0

5

10

15

20

25

0 5 10 15 20 25θu,pred (%)

θ u,e

xp (%

)



(a)

0

5

10

15

20

25

0 5 10 15 20 25θu,pred (%)

θ u,e

xp (%

)



(b)

Fig. 8.4 Experimental ultimate chord rotation of FRP-wrapped columns with continuous bars v

predictions of empirical model of Eqs. (2.13), (4.30b) and (a) (8.8); (b) (8.9).

8.3 FRP-wrapped columns with ribbed (deformed) vertical bars lap-spliced in the plastic

hinge region

All rules proposed in the present section have been developed and calibrated on the basis of

members with FRP wrapping applied over a length exceeding that of the lap. Accordingly, they

should be applied only when such wrapping extends over a length from the end of the member at

least, e.g., 125% of the lapping.

The available tests on rectangular RC members with ribbed (deformed) longitudinal bars

lapped starting at the section of maximum moment show that, in the calculation of the yield

curvature, φy, (used in the 1st and the 3rd term in Eq. (2.13) for θy), as well as of the yield moment,

My, and of the plastic part of the flexure-controlled ultimate chord rotation, θupl, both bars in a pair

of lapped compressed bars should count in the compression reinforcement ratio. Moreover, if the

straight lap length, lo, is less than a minimum value loy,min, then φy and My should be calculated

using as yield stress of the tension reinforcement the value of fy multiplied by lo/loy,min, while the

2nd term of the right-hand-side of Eq. (2.13) should be multiplied by the ratio of the value of My as

modified for the effect of lapping, to its value without it. If the length of the member where the lap

59

splicing takes place is fully wrapped by FRP, which in the presence for FRP, the value of loy,min is:

loy,min = 0.2dbLfy/√fc (fy, fc in MPa) (8.10)

This rule is a modification of Eq. (3.3), derived from members with lap-spliced bars but no

FRP wrapping. Experimental values of My and of the effective stiffness from Eq. (2.14) for FRP-

wrapped columns with lap splices are compared in Fig. 8.5 to predictions with the effect of bar

lapping taken into account according to the above rule. Rows 2, 5 and 8 in Table 8.1 (at the end of

Section 8) refer to the test-to-prediction ratio of My, θy and effective stiffness of such columns.

0

250

500

750

1000

1250

1500

1750

2000

0 250 500 750 1000 1250 1500 1750 2000

M y,e

xp(k

Nm

)

My,pred (kNm)


(a)

0

50

100

150

200

250

0 50 100 150 200 250

EIex

p(M

Nm

2 )

EIpred (MNm2)

median: EIexp=1.045EIpred

(b)

Fig. 8.5 Experimental: (a) yield moment and (b) effective stiffness of FRP-wrapped rectangular

columns with lap-spliced bars, compared to predictions from first principles and Eq. (2.14),

accounting for bar lap-splicing according to Sect. 8.3

Regarding the ultimate chord rotation of members with FRP wrapping of their lap-splice

length, it has been proposed in [8] and adopted in [11] to extend a rule fitted to a large number of

test results on rectangular columns with lap splices confined by the transverse reinforcement

alone. According to this proposal the value of θupl from Eqs. (4.30b), (8.7) is modified as follows,

with lou,min from Eq. (8.12):

θupl = (lo/lou,min)θu

plEqs. (4.30b),(8.7) , if lo < lou,min (8.11)

60

At the same time, if lo is shorter than the value, loy,min, from Eq. (8.10), the first paragraph of the

present section is applied for the effect of bar splicing on θy, to be added to θupl from Eq. (8.11).

In [8] and [11] the minimum lap length beyond which the lapping does not adversely affect the

flexure-controlled ultimate deformation is:

cc

efff

tot

yLbLou

fff

an

fdl

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=,

min,

..ρ4514051

(fyL, ff,e, ,fc in MPa) (8.12)

where ρf, af and ff,e were defined in conjunction with Eqs. (8.1), (8.2), (8.7) and ntot is the total

number of lapped longitudinal bars along the cross-section perimeter (the term 4/ntot is the fraction

of the total number of lap splices confined by the FRP, as in rectangular columns only the four

corner bars, are confined by the FRP wrapped around the corner). Fig. 8.6 compares predictions

with test results. Recent test results show that the value of θu is slightly overestimated following

this approach. The predictions are here compared with 43 experimental results of θu. The mean

value of experimental/predicted ratio is equal to 0.94.

Using a database of the 43 experimental results, it was here re-evaluated the correlation of the

various parameters affecting the ultimate chord rotation of an FRP-wrapped column lap-spliced in

the plastic hinge region. The minimum lap length beyond which the lapping does not adversely affect the

flexure-controlled ultimate deformation, lou,min, is rather affected by (4/ntot)2 instead of 4/ntot that was

suggested by Eq. (8.12). Following this, it is here proposed the following improvement of Eq.

(8.12):

cefffc

u

tot

yLbLou

fff

an

fdl

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

,

min,

..ρ

24514051

(fyL, fu, ,fc in MPa) (8.13)

with (aρfu/fc)f,eff from Eq. (8.8). Evaluation of the 43 test results suggests also that in the case

where 4/ntot < 2/3, the FRP must be considered to improve the clamping of lap-splice, as suggested

by Eq. (8.13), but not to contribute in the enhancement of θupl as calculated by Eq. (4.30b).

According to this, the value of θupl to be used in right-hand side of Eq. (8.11), must be calculated

61

by Eq. (4.30b) without adding the contribution of the FRP to the exponent of the 2nd term from the

end of Eq. (4.30b). In the opposite case where 4/ntot ≥ 2/3, the term of the contribution of the FRP,

Eq. (8.8), should be added to the exponent, in the same way it is suggested in [8] and [11] in

conjunction with Eqs. (4.30b), (8.7), (8.11), (8.12) where the term afρfff,e is added regardless the

value of 4/ntot.

The statistics in rows 22 to 23 of Table 8.1 (at the end of Section 8) suggest that, the here

proposed model for calculating θu of an FRP-wrapped column with lap-splice in the plastic hinge

region, Eqs. (2.13), (4.30b), (8.8), (8.10), (8.11), (8.13), improves the accuracy of the prediction of

θu. Fig. 8.6 compares the predicted values to the experimental results.

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

θ u,e

xp(%

)

θu,pred (%)


Fig. 8.6 Experimental ultimate chord rotation of FRP-wrapped rectangular columns with lap-

spliced bars, compared to predictions from Eqs. (2.13), (4.30b), (8.8), (8.10), (8.11), (8.13).

8.4 Cyclic shear resistance of FRP-wrapped columns

It has been proposed in [8] and adopted in [11] to modify Eq. (6.1b) for the contribution of FRP

wrapping to the cyclic shear resistance of the plastic hinge, as:

( ) ( )( )

fufwfwccs

tot

plcc

sFRPR

zEbVAfhL

fANL

xhV

,

,

5.0;5min16.01)100;5.0max(16.0

;5min05.0155.0;min2

ερρ

μθ

+⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−

⋅−+−

= (8.14a)

62

where Ef and εu,f are the FRP’s Elastic modulus and nominal failure strain and the factor 0.5

accounts for the linear reduction of the FRP stress over the section depth, from its full failure

value of Efεu,f at the extreme tension fibre to zero at the neutral axis. For 10 tests of FRP-wrapped

columns that failed by diagonal tension under cyclic loading after flexural yielding, Fig. 8.7

depicts the test-to-prediction ratio for Eq. (8.14a), as a function of the chord rotation ductility

factor, μθ=θ /θy. Row 24 in Table 8.1 (at the end of Section 8) gives statistics of the ratio of

experimental-to-predicted resistance in diagonal tension, VR.

For consistency with Eq. (8.3) and the effective, average strength of the FRP all around the

column, fu,f = Ef(keffεu,f) [15], [16], as well as owing to a slight downwards tendency of the data in

Fig. 8.7(a), the following alternative is depicted in Fig. 8.7(b):

( ) ( )( )

⎥⎦

⎤⎢⎣

⎡++⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−

⋅−+−

=

fuwfwccs

tot

plcc

sFRPR

zfbVAfhL

fANL

xhV

,

,

;5min16.01)100;5.0max(16.0

;5min05.0155.0;min2

ρρ

μθ

(8.14b)

It is clear from Fig. 8.8 and the statistics at the 2nd row from the bottom of Table 8.1, that the

improvement effected by Eq. (8.14b) is insignificant.

0

0.25

0.5

0.75

1

1.25

1.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5ductility

Vu,e

xp / V

u,pre

d

0

0.25

0.5

0.75

1

1.25

1.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5ductility

Vu,e

xp / V

u,pre

d

(a) (b)

Fig. 8.7 Ratio of experimental shear resistance of FRP-wrapped columns failing by diagonal

tension after flexural yielding to VR from: (a) Eq. (8.14a), (b) Eq. (8.14b).

63

Table 8.1 Mean*, median* and Coefficient of Variation of test-to-prediction ratios for FRP-wrapped rectangular columns with continuous or lap-spliced vertical bars.

Quantity no tests mean* median* Coefficient of variation

1 My,exp/My,pred.-1st-principles continuous bars 188 1.065 1.065 19.3% 2 My,exp/My,pred.-1st-principles lap-spliced bars 44 1.08 1.085 10.8% 3 θy,exp/θy,Eq. (2.13) no pre-damage, continuous bars 144 1.07 0.965 37.7% 4 θy,exp/θy,Eq. (2.13) pre-damaged members, continuous bars 20 1.555 1.455 29.1% 5 θy,exp/θy,Eq. (2.13) no pre-damage, lap-spliced bars 44 1.06 1.05 17.3% 6 (MyLs/3θy)exp/(EIeff)Eqs(2.13),(2.14) no pre-damage, continuous bars 144 1.05 1.03 28.0% 7 (MyLs/3θy)exp/(EIeff)Eqs(2.13),(2.14) pre-damage, continuous bars 20 0.70 0.66 22.1% 8 (MyLs/3θy)exp/(EIeff)Eqs(2.13),(2.14) no pre-damage, lap-spliced bars 44 1.05 1.045 19.7% 9 φu,exp/φu,Eqs(8.1)-(8.3),(8.5),(8.6) continuous bars 33 1.04 1.01 27.5% 10 θu,exp/θu,Eqs(2.13),(8.1)-(8.6) no pre-damage, continuous bars 99 1.075 0.995 37.7% 11 θu,exp/θu,Eqs(2.13),(4.30b),(8.7) no pre-damage, continuous bars 99 1.125 1.09 31.6% 12 θu,exp/θu,Eqs(2.13),(4.30b),(8.8) no pre-damage, continuous bars 99 1.07 1.05 31.2% 13 θu,exp/θu,Eqs(2.13),(4.30b),(8.9) no pre-damage, continuous bars 99 1.06 1.02 31.3% 14 θu,exp/θu,Eqs(2.13),(8.1)-(8.6) pre-damaged continuous bars 18 0.995 0.985 23.1% 15 θu,exp/θu,Eqs(2.13),(4.30b),(8.7) pre-damaged continuous bars 18 0.96 0.93 23.1% 16 θu,exp/θu,Eqs(2.13), (4.30b),(8.8) pre-damaged continuous bars 18 0.94 0.925 23.2% 17 θu,exp/θu,Eqs(2.13), (4.30b),(8.9) pre-damaged continuous bars 18 0.925 0.935 25.4% 18 θu,exp/θu,Eqs(2.13),(8.1)-(8.6) continuous bars, all 117 1.065 0.995 33.4% 19 θu,exp/θu,Eqs(2.13),(4.30b),(8.7) continuous bars, all 117 1.10 1.08 31.2% 20 θu,exp/θu,Eqs(2.13), (4.30b),(8.8) continuous bars, all 117 1.05 1.025 30.6% 21 θu,exp/θu,Eqs(2.13), (4.30b),(8.9) continuous bars, all 117 1.04 1.01 31.0% 22 θu,exp/θu,Eqs. (2.13), (4.30b),(8.7),(8.10)-(8.12) no pre-damage, lap-spliced bars 43 0.91 0.94 35.6% 23 θu,exp/θu,Eqs. .(2.13), (4.30b),(8.8),(8.10),(8.11),(8.13) no pre-damage, spliced bars 43 1.06 1.03 20.4% 24 VR,exp/VR,Eq. (6.1b),(8.14a) diagonal tension failure 10 1.01 1.045 12.9% 25 VR,exp/VR,Eq. (6.1b),(8.14b) diagonal tension failure 10 0.99 1.025 14.1% * For large sample size the median reflects better the average trend than the mean.

64

9 Repaired concrete members

An additional dataset of 33 concrete members (18 rectangular columns and 15 rectangular walls)

in the database includes test results on specimens repaired after testing and re-tested. Few (just

six) had been tested beyond yielding but with little damage, few others (five) had suffered more

serious damage, while most (22) were tested beyond conventionally defined ultimate deformation

(20% post-ultimate strength drop of lateral force resistance). Specimens were then repaired to

restore their original lateral resistance, effective stiffness and deformation capacity. The repair

methods used included epoxy grouting of cracks (in just 6 specimens), replacement with non-

shrink mortar of disintegrated cover concrete (7 specimens) or cover and core concrete (9

specimens), cutting of buckled longitudinal bars and replacement with welding (9 specimens), etc.

The expressions in Sections 2 and 4 may be applied to the repaired member, assuming that the

strength of the repair concrete used in the plastic hinge (typically higher than that of the original

concrete) applies to the whole element. The test-to-prediction ratio for the yield moment, My, the

chord rotation at yielding, θy, the secant stiffness to the yield-point and (for the repaired specimens

carried to flexural failure), θu, have means, medians and coefficients of variation shown in Table

9.1.

Although based on limited data, the comparisons in Table 9.1 show that, even when carried

out as carefully as in a research lab, repair re-instates fully only the yield moment (and hence the

moment resistance), failing by 25-30% to recover the secant stiffness to the yield-point and the

deformation capacity. Interestingly, repaired walls exhibit much larger loss of stiffness than

repaired columns, but they fare a little better than columns at ultimate (although the difference is

statistically insignificant). Although the small sample size normally reduces the apparent scatter,

the dispersion of test results with respect to predictions is much larger than in virgin specimens,

even for the yield moment which is recovered well on average. Apparently, not only the repair

process and materials, but also the type and degree of the original damage, introduce significant

additional uncertainty.

65

Table 9.1: Mean*, median* and coefficient of variation of test-to-prediction ratios for repaired

concrete members

Test-to-predicted ratio No of data

mean* median* Coefficient of variation

1 My,exp/My,pred.-1st-principles columns with rectangular section 18 0.925 1.005 25.8% 2 My,exp/My,pred.-1st-principles walls (all with rectangular section) 15 1.045 1.035 26% 3 My,exp/My,pred.-1st-principles all tests 33 0.98 1.015 26.2% 4 θy,exp/θy,Eq. (2.13a) columns with rectangular section 18 1.215 1.265 23.8% 5 θy,exp/θy,Eq. (2.13b) walls (all with rectangular section) 15 1.65 1.665 40.7% 6 θy,exp/θy,Eq. (2.13b) all tests 33 1.41 1.27 38% 7 (MyLs/3θy)exp/(EIeff)Eqs(2.13),(2.14) columns with rectangular section 18 0.78 0.79 32.4% 8 (MyLs/3θy)exp/(EIeff)Eqs(2.13),(2.14) walls (all with rectangular section) 15 0.745 0.535 58.8% 9 (MyLs/3θy)exp/(EIeff)Eqs(2.13),(2.14) all tests 33 0.765 0.725 45%

10 θu,exp/θu,Eqs. (4.28),(4.29) columns with rectangular section 15 0.86 0.705 54% 11 θu,exp/θu,Eqs. (4.28),(4.29) walls (all with rectangular section) 15 1.10 0.97 58.6% 12 θu,exp/θu,Eqs. (4.28),(4.29) all tests 30 0.98 0.805 57.7% 13 θu,exp/θu,Eqs. (4.30a) columns with rectangular section 15 0.725 0.675 32.6% 14 θu,exp/θu,Eqs. (4.30a) walls (all with rectangular section) 15 0.61 0.70 55% 15 θu,exp/θu,Eqs. (4.30a) all tests 30 0.67 0.69 43.6% 16 θu,exp/θu,Eqs. (4.30b) columns with rectangular section 15 0.755 0.72 32.4% 17 θu,exp/θu,Eqs. (4.30b) walls (all with rectangular section) 15 0.62 0.695 58.6% 18 θu,exp/θu,Eqs. (4.30b) all tests 30 0.69 0.71 45.4% 19 θu,exp/θu,Eqs. (4.30c) columns with rectangular section 15 0.74 0.705 32.6% 20 θu,exp/θu,Eqs. (4.30c) walls (all with rectangular section) 15 0.735 0.825 52.8% 21 θu,exp/θu,Eqs. (4.30c) all tests 30 0.735 0.74 43.1%

* For large sample size the median reflects better the average trend than the mean.

67

REFERENCES

[1] ASCE. Prestandard for the seismic rehabilitation of buildings. Prepared by the American

Society of Civil Engineers for the Federal Emergency Management Agency (FEMA Report

356), Reston, VA, 2000.

[2] ASCE/SEI Standard 41-06 “Seismic rehabilitation of existing buildings”. American Society of

Civil Engineers, Reston, VA, 2007.

[3] Ascheim, MA Moehle JP "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, 1992.

[4] Benjamin JR, Cornell CA Probability, Statistics and Decision for Civil Engineers. McGraw

Hill, New York, N.Y., 1980.

[5] Bigaj A, Walraven JC “Size effect on rotational capacity of plastic hinges in reinforced concrete

beams”. CEB Bulletin d' Information No 218, Lausanne, 1993, pp. 7-23.

[6] Biskinis DE, Fardis MN “Cyclic Strength and Deformation Capacity of RC Members,

Including Members Retrofitted for Earthquake Resistance”, 5th International Ph.D Symposium

in Civil Engineering, Delft (NL), 2004, Balkema: 1125-1133.

[7] Biskinis DE, Roupakias, G, Fardis MN “Degradation of Shear Strength of RC Members with

Inelastic Cyclic Displacements”, ACI Structural Journal, vol. 101, No. 6, Nov.-Dec. 2004, pp.

773-783.

[8] Biskinis DE “Resistance and Deformation Capacity of Concrete Members with or without

Retrofitting”. Doctoral Thesis, Civil Engineering Department, University of Patras, Greece,

2007.

[9] Bousias SN, Biskinis DE, Fardis MN, Spathis L-A (2007) Strength, Stiffness and Cyclic

Deformation Capacity of Concrete Jacketed Members, ACI Structural Journal, vol. 104, No. 5,

Sept.-Oct. 2007, pp. 521-531.

[10] Eligehausen R, Lettow S “Formulation of Application Rules for Lap Splices in the New fib

68

Model Code”, Stuttgart, 2007.

[11] European Standard EN1998-3:2005 “Eurocode 8: Design of structures for earthquake

resistance, Part 3: Assessment and retrofitting of buildings”, Comite Europeen de

Normalisation, Brusells, 2005.

[12] European Standard EN1992-1-1: “Eurocode 2 - Design of Concrete Structures - Part 1-1:

General Rules and Rules for Buildings”, Comite Europeen de Normalisation, 2004, Brusells.

[13] fib SAG5. “New Model Code – First Draft”, fib, Lausanne, 2009.

[14] Kowalsky M Priestley MJN "Improved Analytical Model for Shear Strength of Circular

Reinforced Concrete Columns in Seismic Regions" Structural Journal, American Concrete

Institute, Vol. 97, No. 3, 2000, pp.388-396.

[15] Lam L, Teng JG “Design-oriented Stress-Strain Modle for FRP-Confined Concrete”,

Construction and Building Materials, 17, 6 & 7, 2003, pp. 471-489.

[16] Lam L, Teng JG “Design-oriented Stress-Strain Model for FRP-Confined Concrete in

Rectangular Columns”, Journal of Reinforcing Plastics and Composites, 22, 13, 2003,

pp.1149-1186

[17] Moehle J, Lynn A, Elwood K, Sezen H "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, 2001.

[18] Newman K, Newman JB “Failure Theories and Design Criteria for Smooth Concrete”, in

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[19] Panagiotakos TB, Fardis MN “Deformations of RC Members at Yielding and Ultimate”,

ACI Structural Journal, Vol. 98, No. 2, 2001, pp.135-148.

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Champaign, ILL, 1928.

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71

NOTATION

a: clear distance between anchored bars or pairs of lapped bars

acy: zero-one variable for cyclic or monotonic loading for the ultimate chord rotation

αeff,j: effectiveness factor of the FRP jacket

aEI: coefficient in empirical expression for EIeff

al,s: effectiveness factor of transverse reinforcement in confining lapped bars

As,tot: total cross-sectional area of longitudinal reinforcement

asl: zero-one variable for slip of longitudinal bars from their anchorage zone beyond end section

ast, aplst, ahbw

st: coefficients for the type of steel in Eqs. (4.30) for the ultimate chord rotation

av: zero-one variable for diagonal cracking before flexural yielding of the end section

aw,r: zero-one variable for rectangular walls in Eqs. (4.30) for the ultimate chord rotation

aw,nr: zero-one variable for non-rectangular sections in Eqs. (4.30) for ultimate chord rotation

Ac: gross cross-sectional area of concrete member

b: width of compression zone

bo: width of confined concrete core to the hoop centreline

bi: centreline spacing along the section perimeter of longitudinal bars (indexed by i) engaged by a

stirrup corner or a cross-tie

bx and by: the cross-sectional dimensions of the section

bw: web width of the cross-section (both webs in U- or hollow rectangular sections)

c: clear cover of longitudinal bars

d: effective depth of cross-section

do: effective depth of confined concrete core to hoop centreline

d’: distance of the center of compression reinforcement from extreme compression fibres

dbL: diameter of longitudinal reinforcement

Ec: Elastic modulus of concrete

Ef: Elastic modulus of the FRP

72

Es: Elastic modulus of steel

EIeff: member effective flexural rigidity, taken as its secant stiffness to yielding in a bilinear force-

deformation model

EIc: uncracked gross section stiffness

fc: unconfined compressive strength of concrete based on standard cylinder test

fcc: compressive strength of confined concrete

fsm: maximum possible tensile stress a lap-spliced bar can develop

ft: ultimate strength of reinforcing steel

fu,f : effective strength of the FRP

ff,e: effective stress of the FRP

ffu,nom: nominal strength of the FRP

fy: yield stress of longitudinal reinforcement (subscript 1 is for tension bars)

fyw: yield stress of transverse reinforcement

h: depth of cross section

ho: depth of confined concrete core to the hoop centreline

k: effectiveness factor of transverse bars for clamping anchored or lap-spliced bars

keff: FRP effectiveness factor, Eq. (8.3)

lgauge : gauge length over which relative rotations are measured

lo: lap length of longitudinal bars

loy,min: minimum required lap length for the member to develop its full yield moment

lou,min: minimum required lap length for a member to develop its full ultimate deformation as if its

bars were continuous

Lpl: plastic hinge length

Ls: shear span (=M/V at the member end)

Ls/h=M/Vh: shear span ratio at member end

ΜR: moment resistance of RC section.

73

ΜRo: moment resistance of confined core of spalled section

ΜRc: moment resistance of unconfined full (unspalled) section

My: yield moment of cross section

My*: yield moment of jacketed member

My,exp: moment at the corner of a bilinear envelope of the monotonic or cyclic experimental force-

deformation response

My,pred: value of yield moment from plane section analysis with linear σ-ε material laws

nrestr: number of lapped longitudinal bars engaged by a stirrup corner or a cross-tie

ntot: total number of lapped longitudinal bars along the perimeter of the cross-section

Nb,tension: number of bars in the tension zone

N: axial force (positive for compression)

N1, N2: values of N delimiting brittle shear-controlled failure of squat column.

p: active confining pressure normal to axis of developed or lapped bar due to external loading

sh: spacing of transverse reinforcement

t: thickness of compression flange in a T-, L-, H- or U-section

VRc: shear force at diagonal cracking

x: compression zone depth;

z: internal lever arm

α=Es/Ec: ratio of steel to concrete moduli

α: confinement effectiveness factor

αf: effectiveness factor for confinement by FRP of a rectangular section having its corners rounded

to a radius R to apply the FRP

δ’=d’/d

Δθu,slip: fixed-end rotation due to bar slippage from the anchorage zone beyond the end section,

that takes place between yielding and the flexure-controlled ultimate deformation

εc: strain at extreme compression fibre, beyond which section is considered to “yield” due to

74

concrete in compression

εcu: ultimate strain of unconfined concrete

εcu,c: ultimate strain of confined concrete.

εy: yield strain of steel reinforcement

εsh: steel strain at outset of strain-hardening

εsu: elongation of continuous steel bars when ultimate curvature is reached by steel rupture

εsu,cy: elongation of continuous steel bars when ultimate curvature under cyclic loading takes place

by steel rupture

εsu,mon: elongation of continuous steel bars when ultimate curvature under monotonic loading takes

place by steel rupture

εsu,l: elongation of lapped steel bars when ultimate curvature takes place by steel rupture in a

member with bars lap-spliced in the plastic hinge region

εsu,nominal: elongation at tensile strength in standard monotonic test of steel coupons

εu,f: failure strain of the FRP

θ: chord rotation at a member end (angle between the normal to the end section and the chord

connecting the member ends at the member’s displaced position)

θslip: fixed-end rotation due to bar slippage from anchorage zone beyond end section

θu: flexure-controlled ultimate chord rotation (at 20% post-ultimate strength drop in lateral force

resistance).

θu*: ultimate chord rotation (drift ratio) of jacketed member

θupl: plastic part of the ultimate chord rotation

θu,exp: experimental value of ultimate chord rotation

θu,pred: predicted value of θu

θuy: y-axis component of the experimental ultimate chord rotation under biaxial loading

θuz: z-axis component of the experimental ultimate chord rotation under biaxial loading

75

θuy,uni: predicted value of θu for uniaxial loading along the y-axis of the member’s section

θuz,uni: predicted value of θu for uniaxial loading along the z-axis of the member’s section

θplu: plastic part of θu in a member with continuous bars

θplu,l: plastic part of θu in a member with bars lap-spliced in the plastic hinge region

θy: chord rotation at the corner of a bilinear envelope of the monotonic or cyclic force-deformation

response (at “yielding”)

θy*: chord rotation (drift ratio) of jacketed member at yielding

θy,exp: experimental value of θy

θy,pred: predicted value of θy

θy,slip: value of θslip at member “yielding”

θyy,exp: experimental chord rotation along member y-axis at yielding under biaxial loading

θyz,exp: experimental chord rotation along member z-axis at yielding under biaxial loading

θyy,uni: predicted value of θy for uniaxial loading along member y-axis

θyz,uni: predicted value of θy for uniaxial loading along member z-axis

μθ=θ /θy: chord rotation ductility factor

ν: normalized axial load, N/bhfc

ξ: neutral axis depth of cross-section, normalized to d

ξu: value of ξ at ultimate curvature of the section

ξy: value of ξ at yielding of the section

ρ1=As1/bd: tension reinforcement ratio

ρ2=As2/bd: compression reinforcement ratio

ρd: diagonal reinforcement ratio in diagonally reinforced members (ratio of cross-sectional area of

reinforcement along one diagonal to bd)

ρf: geometric ratio of the FRP parallel to the direction of bending

ρs: minimum transverse reinforcement ratio among the two transverse directions

76

ρtot=ρ1+ρ2+ρv: total longitudinal reinforcement ratio

ρv=Asv/bd: ratio of “web” longitudinal reinforcement , uniformly distributed between tension and

compression reinforcement

ρw=Ash/bsh: transverse reinforcement ratio

σs: stress of tension bars at the end section of the member

φ: section curvature

φu: section ultimate curvature (at 20% post-ultimate strength drop in lateral force resistance)

φcu: value of φu reached by failure of the compression zone

φsu: value of φu reached by fracture of the tension reinforcement

φy: section curvature at yielding

ω1: mechanical reinforcement ratio of tension longitudinal reinforcement, ρ1fy/fc

ω2: mechanical reinforcement ratio of compression reinforcement, ρ2fy/fc

ωtot=ρtotfy/fc : total mechanical reinforcement ratio

ωw=ρwfyw/fc : mechanical transverse reinforcement ratio

deformations of concrete members at ......over 300 uniaxial tests are identified to fail in shear...

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