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Content may be subject to copyright.

131

© Institution of Engineers Australia, 2014

* Paper S13-002 submitted 11/01/13; accepted for publication

after review and revision 6/03/13.

†

Corresponding author Prof John Wilson can be contacted at

jwilson@swin.edu.au.

technical paper

Drift capacity of lightly reinforced concrete columns

*

A Wibowo and JL Wilson

†

Faculty of Engineering and Industrial Science, Swinburne University of Technology, Hawthorn, Victoria

NTK Lam

Department of Civil & Environmental Engineering, University of Melbourne, Parkville, Victoria

EF Gad

Faculty of Engineering and Industrial Science, Swinburne University of Technology, Hawthorn, Victoria

ABSTRACT: This paper presents the ndings of a research project investigating the lateral

load drift behaviour of lightly reinforced concrete columns. Such columns of limited ductility are

common in regions of low-moderate seismicity, and although their strength properties are well

de ned, the drift performance is less understood. The paper presents the results of an experimental

study undertaken and the development of a theoretical model for predicting the lateral load-drift

behaviour of lightly reinforced concrete columns together with a simpli ed bi-linear model for

checking purposes. The test results are presented and clearly indicate the dramatic impact that

the axial load ratio has on the drift performance of columns of limited ductility, particularly the

signi cantly lower drift capacities that are available in compression dominated columns.

KEYWORDS: Drift capacity; axial load ratio; reinforced column tests; limited ductility;

columns; seismic performance.

REFERENCE: Wibowo, A., Wilson, J. L., Lam, N. T. K. & Gad, E. F. 2014, “Drift capacity

of lightly reinforced concrete columns”, Australian Journal of Structural Engineering, Vol.

15, No. 2, April, pp. 131-150, http://dx.doi.org/10.7158/S13-002.2014.15.2.

1 INTRODUCTION

Lightly reinforced concrete columns are prevalent

in many old buildings and common in current

detailing practice in the regions of lower seismicity.

This type of structure is believed to have a very low

lateral load and drift capacity from a conventional

design perspective. However, many post-earthquake

investigations have shown that the primary cause

of reinforced concrete building collapse during

earthquakes is the loss of vertical-load-carrying

capacity in critical building components leading to

catastrophic vertical collapse, rather than a reduction

in the lateral-load capacity (Otani, 1997; Wibowo et

al, 2008; Moehle et al, 2002).

The capacity spectrum method (ATC40) (Wilson

& Lam, 2006) provides a very convenient method

for assessing the seismic performance of structural

system by superimposing the structural capacity

curve (push-over curve) with the seismic demand

curve expressed in the form of an acceleration-

displacement response spectrum (ADRS) as shown

in figure 1 (where RSA, RSV and RSD refer to

the response spectral acceleration, velocity and

displacement values, respectively).

The ADRS demand curve can be obtained from

the relevant seismic design code for the region,

eg.Australia, AS1170.4 (Standards Australia, 2007). In

contrast, the push-over curve is structure dependent

and requires knowledge of the relationship between

the lateral force and the associated drift in both the

elastic and inelastic range. The push-over curve for

well detailed ductile columns has been thoroughly

researched and documented (Priestley et al, 1996;

Australian Journal of Structural Engineering, Vol 15 No 2

132

Australian Journal of Structural Engineering

Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

2007), with many design guidelines, such as

ATC40 (ATC, 1996) available. The lateral strength

properties of lightly reinforced concrete columns

are well understood and codi ed, but in contrast,

the associated drift estimates beyond peak strength

are less well de ned making it dif cult to construct

a realistic push-over curve. Further, many design

guidelines, such as FEMA273 (FEMA, 1997) and

ATC40, apply very conservative drift limits to lightly

reinforced concrete columns (Wilson et al, 2009;

Elwood & Moehle, 2003).

The overall aim of this paper is to investigate the

load-de ection behaviour and drift capacity of lightly

reinforced concrete columns, which many design

guidelines assume to be brittle in nature with very

low drift capacities, and to develop a simpli ed

push-over curve that can be used to assess the seismic

performance of such structures. An experimental

program undertaken by the authors involving

four column specimens is described in section 2,

while the overall results and speci c displacement

measurements are presented in sections 3 and 4,

respectively. Theoretical displacement estimates are

presented in section 5 together with a simpli ed code

bi-linear model for assessing the drift at maximum

strength presented in section 6. Finally, conclusions

are presented in section 7.

2 EXPERIMENTAL TEST SETUP

Four column specimens were designed to represent

the non-ductile detailed reinforced concrete columns

of old buildings commonly found in low-to-moderate

seismic regions (Wibowo et al, 2010a; 2010b; Wilson

et al, 2009). The two parameters varied were the axial

load ratio (n) and the longitudinal steel reinforcement

ratio (

v

), while both the transverse reinforcement

ratio (

h

) and the aspect ratio (a) remained constant.

(The axial load ratio (n) is de ned as the ratio of

the applied axial stress to the concrete crushing

strength, while the aspect ratio (a) is defined as

the column height to width ratio.) The cantilever

column specimens measured 300×270×1200 mm

high with an aspect ratio of a = 4. Column specimens

S1-S2 were loaded to create an axial load ratio

n = 0.20, while specimens S3-S4 were loaded to

n= 0.40. The longitudinal steel ratio of specimens

S1 and S4 was

v

= 0.56% (4N12 reinforcing bars),

which was below the minimum threshold level of

reinforcing allowed in AS3600 (Standards Australia,

2001) of 1.0%, while specimens S2 and S3 had the

minimum reinforcement ratio of

v

= 1.0% (4N16

reinforcing bars). The transverse steel ratio for all

specimens was

h

= 0.07% (area ratio), consisting

of R6 at 300mm stirrups which was less than the

minimum lateral reinforcement required by AS3600

of 0.09%. All stirrups had 135° hooks but with only

half the hook length of current design codes. The

low reinforcement ratios were deliberately selected

to investigate whether such columns behaved in a

brittle fashion as assumed in many design guidelines

(such as FEMA273 and ATC40) or had some level of

ductility and drift capacity when subjected to lateral

loading. The concrete cover was 20 mm, while the

speci ed concrete compressive strength was 20 MPa

and the ductile steel yield stresses were 536 MPa

for the main reinforcement and 362 MPa for the

stirrups. A summary of the four column specimens

are presented in gure 2 and table 1.

The drift capacity of concrete columns is made up

of exural, yield penetration, and shear components

which were all measured using linear variable

displacement transducers (LVDTs) and strain

gauges. The axial displacement was also measured

to detect any loss of axial-load carrying capacity.

Displacements were measured using 18 LVDTs, as

shown in gure 3(a). The LVDTs were arranged

to measure the axial displacement (no. 18), total

lateral displacement (no. 1-5), exural displacement

(no.6-11) and shear deformation (no. 12-17), while 16

strain gauges were installed on the reinforcement to

measure the longitudinal and transverse strains as

shown in gure 3(b). Three strain gauge locations

Fig ure 1: Capacity spectrum method.

133

Australian Journal of Structural Engineering Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

were used; one level for checking yield penetration

length, the second level at the footing-column surface

for measuring the maximum strain and estimating

the yield penetration; and the third level was in the

middle of predicted plastic hinge length.

The axial load was applied and maintained using

a hydraulic jack, while the lateral load was applied

using a hydraulic actuator with a 100 ton loading

capacity as shown in gure 4. The displacement

controlled loading sequence consisted of drift-

controlled mode at drift increments of 0.25% until

reaching 2% drift, and then followed by drift

increments of 0.5% (where the drift is de ned as the

ratio of the lateral displacement to column height

expressed as a percentage). Two cycles of loading

were applied at each drift ratio to ensure that the

hysteretic behaviour could be maintained. The lateral

loading was held constant at various stages while the

LVDT and strain gauge measurements were taken,

crack patterns recorded, and visual inspections made.

The test ended when the column lost the capacity

to resist and support to the axial load (axial failure)

rather than the more traditional failure de nition

of the peak lateral loading capacity of the specimen

reducing by 20% (lateral load failure).

3 OVERALL EXPERIMENTAL RESULTS

Specimen S1 with

v

= 0.56% rebar ratio and n = 0.20

axial load ratio was able to sustain a maximum drift

of 5.0% prior to axial load collapse with classical

plastic hinge formation at the base of the column and

a rigid body rocking mechanism as shown in gure

5. Such desirable behaviour is associated with yield

Fig ure 2: Geometry and reinforcement details of column specimens S1-S4.

Table 1: Basic properties of column specimens.

Spec

Dimension

[mm]

L

[mm]

L/D

ρ

v

[%]

Main

rebars

ρ

h

(%)

Stirrups

[@mm]

n

f

c

’

[MPa]

Hook

type

Area Volumetric

S1 270×300×1200 1200 4 0.56 4N12 0.07 0.10 R6@300 0.2 20.3 135°

S2 270×300×1200 1200 4 1.00 4N16 0.07 0.10 R6@300 0.2 21.0 135°

S3 270×300×1200 1200 4 1.00 4N16 0.07 0.10 R6@300 0.4 18.4 135°

S4 270×300×1200 1200 4 0.56 4N12 0.07 0.10 R6@300 0.4 23.7 135°

Notation: L is shear span which is the clear-height of the column; L/D is the aspect ratio de ned as shear span divided by

the column depth; n is the axial load ratio (ratio of the axial load to axial load-carrying capacity A

g

f

c

’);

v

is the longitudinal

reinforcement ratio (

v

= A

s

/A

g

);

h

is the lateral reinforcement (A

sh

/bs); A

sh

= total area of transverse reinforcement; s = tie spacing;

and b = column section width.

134

Australian Journal of Structural Engineering

Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

Fig ure 3: Instrumentation for column specimens S1-S4 – (a) LVDTs and (b) strain gauges.

(a)

(b)

135

Australian Journal of Structural Engineering Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

Fig ure 4: Setup of column specimens S1-S4.

Fig ure 5: Failure damage of column specimens (a) S1, (b) S2, (c) S3 and (d) S4.

(a) (b) (c) (d)

penetration and damage localised at the base rather

than cracking and spalling of the concrete spreading

above the base.

In contrast, specimens S2 and S3 with almost twice

the longitudinal reinforcement

v

= 1.0% tolerated

lower maximum drifts of 2.5% and 1.5% for axial

load ratio of n = 0.20 and n = 0.40, respectively, as

shown in gure 5. The large tie spacing (300 mm) in

these specimens led to buckling of the longitudinal

reinforcement after the concrete cover had spalled

and an abrupt transfer of axial load from the steel

reinforcement to the damaged concrete. This triggered

lateral load failure due to the deterioration of the

concrete under the cyclic loading. Specimens S4 and

S3, both with an axial load ratio of n = 0.40 responded

in a similar fashion with a maximum drift ratio of

1.5%, despite the different longitudinal rebar ratio.

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Australian Journal of Structural Engineering

Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

All specimens exhibitedclassicalreinforced concrete

(RC) column behaviour up to the peak strength,

with lateral load peaks for the same axial load

ratio occurring at a similar drift, but the post peak

deterioration of specimens with

v

= 1.0% rebar

ratio was much faster than that of specimens with

v

= 0.56% rebar ratio. The post peak behaviour of

all specimens except S1 could be predicted using

classical moment-curvature relationships.

The hysteresis curve and backbone moment-drift

curve for each specimen is presented in gures 6

and 7 respectively (the backbone curve was derived

from the hysteresis curve and is the effective pus-over

curve for the column allowing for cyclic deterioration

effects). A summary of the test results is listed in table

2 including the lateral load and drift values at failure

de ned by (i) 80% of the maximum lateral load-

carrying capacity and (ii) gravity axial load collapse.

The first crack and first yield drift values for

specimens S1 and S2 were less than those of the more

heavily loaded specimens S3 and S4. Conversely, the

peak strength of specimens S1 and S2 occurred at

larger drifts than those of specimens S3 and S4. This

phenomenon could be attributed to the higher axial

load ratio of specimens S3 and S4 that increased the

compressive stress of column cross-sections, and

in turn reduced the curvature ductility capacity

through an increase of yield curvature and a decrease

of ultimate curvature. An increase in the axial load

ratio from n = 0.2 to n = 0.4 resulted in an increase

in the ultimate lateral strength capacity by about

30% for columns with main rebar ratio of

v

=0.56%

(specimens S1 and S4), but had little effect for

columns with

v

= 1.0% main rebar ratio (specimens

S2 and S3). In contrast, the effect of an increase

of main rebar ratio from

v

= 0.56% to

v

= 1.0%

Fig ure 6: Hysteretic curves for column specimens S1-S4.

Fig ure 7: Backbone moment-drift curves for column specimens S1-S4.

137

Australian Journal of Structural Engineering Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

increased the ultimate lateral strength capacity of

columns with n = 0.2 axial load ratio by about 30%

(specimens S1 and S2).

An increase of axial load ratio from n = 0.2 to n=0.4

reduced the ultimate drift capacity for column with

v

= 0.56% rebar ratio (specimens S1 and S4) by

about 70% compared with a 40% drift reduction

for the

v

=1.0% rebar ratio specimens (S2 and S3).

The measured maximum moment strengths M

u,exp

represented an over-strength factor in the range

= 1.4-1.7 compared with the factored design

moment strengths (

M

u

) derived from AS3600

for specimens S1-S4 as shown in figure 7 and

summarised in table 3.

The hook length used in the transverse reinforcement

of all specimens was about half of the required

hook length, but interestingly, an opening hook was

barely found in any specimen. Clearly, the stirrup

spacing has a greater effect rather than hook type and

length in lightly reinforced concrete columns, since

the transverse reinforcement tends to help prevent

longitudinal reinforcement buckling and provide

additional shear strength but has little effect on the

concrete con nement.

4 EXPERIMENTAL DISPLACEMENT

MEASUREMENTS

The lateral displacement of a column (

tot

) consists

of three components, exural (

), yield penetration

(

yp

) and shear deformation (

sh

) which are illustrated

in figure 8 and explained from an analytical

perspective in this section.

tot

=

+

yp

+

sh

(1)

4.1 Flexural displacement

Flexural de ection prediction of the member is well

understood and evaluated by performing curvature

Table 2: Experimental results for specimens S1-S4.

Parameters Unit S1 S2 S3 S4

V

cr

KN 24 28 35 39

V

y

KN 51 67 79 76

V

max

KN 59 79 81 77

V

80%

KN 48 63 67 64

V

collapse

KN 18 28 67 64

cr

% drift 0.13 0.13 0.20 0.20

y

% drift 0.75 0.75 1.00 1.00

Vmax

% drift 1.71 1.73 1.12 1.01

80%

% drift 3.30 2.10 1.50 1.50

collapse

% drift 5.00 2.50 1.50 1.50

Drift ductility – 6.7 3.3 1.5 1.5

Table 3: The over-strength factor as a ratio of

the measured ultimate strength to the

nominal factored moment capacity

(AS3600).

Specimen

ϕ

M

u

ϕM

u

M

u.exp

Ω

S1 0.7 68 48 72 1.45

S2 0.7 92 64 94 1.39

S3 0.6 98 59 99 1.69

S4 0.6 90 54 92 1.71

integration over the column height with the curvature

distribution obtained from a bre section model. A

plastic hinge analysis is needed to determine the

inelastic deformation from the regions where the

applied moment exceeds the yield moment, resulting

in large curvatures.

The average curvature (

) has been calculated using

equation (2) over the four segments up the height of

the column as shown in gure 8(a). The three lower

segments which covered the predicted cracked

region were instrumented with LVDTs, while the

average curvature (

) of the upper segment could

be conservatively calculated using elastic properties

since no cracks were observed in this region.

21

1

ff

VhV

LLL

(2)

where L

V

= height per each segment, L

h

= distance

between flexural LVDTs, and

f

= vertical LVDT

measurement.

The exural displacement component at the column

top for each LVDT segment was obtained using:

.21

0

()

L

Vi

fi i f f

H

LL

x xdx

L

(3)

138

Australian Journal of Structural Engineering

Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

D

L

H

d

1

d

2

c

s

lip

E

L

V1

G

f 1

G

f 2

T

slip

d

d’

sg

1

sg

2

'

fl

'

sh

'

sh1

L

L

v4

L

v3

L

v2

L

v1

G

f1

G

f2

E

L

h

G

s1

G

s2

]

D

L

L

v3

L

v4

L

v2

L

v1

(b) Shear (a) Flexure

Fig ure 8: Measurements obtained from transducers and strain gauges – (a) flexure, (b) shear,

and (c) yield penetration.

(a) (b)

(c)

While the displacement of the upper segment

without a LVDT transducer was calculated assuming

uncracked section properties:

23

,4

33

ii

fi

c

LVL

EI

(4)

where V = lateral load, L

i

= segment length, E

c

=

concrete elastic modulus, and I = uncracked second

moment of area.

The resulting total exural displacement at the top

of the column could then be obtained as the sum of

the components:

4

,

1

fi fi i

i

(5)

The measured average curvatures over the height of

the column are shown in gure 9, while the exural

displacement contributions from both within and

139

Australian Journal of Structural Engineering Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

outside the plastic hinge regions are shown in gure

11 for the four column specimens. Figure 9 shows

that only specimen S1 developed a plastic hinge

mechanism within the predicted plastic hinge length,

while the plastic hinge region of the other three

specimens S2-S4 extended to the second stirrup.

The additional exural displacement attributed to

deformations outside the plastic hinge region for

specimens S2-S4 are clearly shown in gure 11.

4.2 Yield penetration displacements

Yield penetration or slip deformation is characterised

by a rigid body rotation of the column associated

with a gap opening at the column-foundation

interface. The reinforcement yields at this location

and the inelastic deformation penetrates into the

foundation due to bond failure. Consequently, the

reinforcement elongates locally, the gap widens and

the column rotates.

The yield penetration effect was obtained from the

strain gauge and LVDT measurements at the column

base interface as shown in gure 8(c) by assuming a

rocking mechanism within the rst section of column.

The slip displacement of the tensile steel at the gap

opening can be calculated as follows:

slip =

sg1

f

s

d

b

/4u (6)

while the shortening displacement of the compressive

steel can be obtained via:

sc

=

sg2

L

column

(7)

where

sg1

and

sg2

are strain gauge reading at tensile

and compressive steel respectively, and u

e

is the bond

stress between concrete and steel (u

e

=

c

f

by Sezen

& Moehle, 2003, is used in this study).

Hence, the neutral axis depth at the column base

interface can be estimated via:

sc

sc

c d d' d'

slip

(8)

While, the slip rotation

slip

is given by:

slip

= slip/(d – c) (9)

An upper bound of the slip displacement can be

determined from the vertical LVDT measurement

L

L

v4

L

v3

L

v2

L

v1

G

f1

L

h

G

f2

G

f3

G

f4

G

f5

G

f6

2

34

2

1

Vh

ff

LL

G

G

M

1

12

1

1

Vh

ff

LL

G

G

M

3

56

3

1

Vh

ff

LL

G

G

M

M

u

M

y

L

p

Fig ure 9: Average curvature calculations for column specimens S1-S4. (a) Idealised tri-linear and bi-linear

shape of measured curvature distribution, and (b) curvature distribution over column height.

(a)

(b)

140

Australian Journal of Structural Engineering

Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

at the rst level (no. 8 and 11). While the neutral

axis depth at the column base interface can be

estimated via:

2

2

12

f

H

ff

cLd

(10)

The slip rotation of the tensile steel can then be

obtained using:

21ff

slip

H

slip

dc L

(11)

Hence, the related slip displacement can be calculated

using:

slip =

(d + d

2

) –

f2

(12)

The top displacement of the column can then be

calculated from the product of the slip rotation

slip

and the column height assuming rigid body rotation.

yp

=

slip

L

column

(13)

The yield penetration contributions to the total

displacement at the top of the columns for specimens

S1-S4 are plotted in figure 11. Reasonable yield

penetration drifts in the order of 0.5% were measured

in specimen S1 due to the low axial load ratio and

low reinforcement ratio, while much smaller drifts in

the order of 0.2% for specimen S2-S4 were measured.

4.3 Shear displacement

The measured shear deformation

sh

was estimated

from the diagonal LVDT measurements (refer gure

8(b)), such that:

22

12 12

sec

22

v

ss ss

sh

v

LD

L

(14)

where

s

= diagonal LVDT measurement and D= cross-

section depth (parallel to lateral loading direction).

The average shear displacement distribution over the

height of the column was calculated using equation

(14) and is presented in gure 10 for all specimens.

It was found that the shear deformation of specimen

S1 with the smallest longitudinal reinforcement

ratio and axial load ratio was mostly concentrated

in the plastic hinge region, while specimens S2, S3,

and S4 developed shear deformations up to the

second stirrup region. A prompt increase of shear

deformation was observed in specimens S2, S3 and

S4 that was attributed to rapid broadening and

propagation of the main diagonal cracks. However,

specimens S3 and S4 with a higher axial load ratio

developed larger shear deformations at an earlier

stage of drift compared with specimen S2. The overall

drift due to shear deformation was small and in the

order of 0.1-0.2% for specimens S1-S4 as shown in

gure 11.

4.4 Total lateral displacement

Figure 11 and table 4 show the various components of

the measured column lateral drifts for all specimens

as a function of lateral load in order to qualitatively

indicate the modes of failure.

5 THEORETICAL DISPLACEMENT

PREDICTIONS

In this section, a detailed theoretical analysis

comprising exural, yield penetration, and shear

displacement is presented and compared with the

experimental results.

5.1 Flexural displacement

The plastic hinge lengths for all specimens were

estimated using a tri-linear representation of the

Fig ure 10: Shear displacement distributions for column specimens S1-S4.

141

Australian Journal of Structural Engineering Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

0

10

20

30

40

50

60

70

80

90

100

00.511.522.533.5

Lateral Load (KN)

Drift (%)

flexural - within Lp flexural - outside Lp shear

yield penetration total displacement

S1 (Uv=0.56% ; n=0.2)

0

10

20

30

40

50

60

70

80

90

100

0 0.5 1 1.5 2 2.5

Lateral Load (KN)

Drift (%)

flexural - within Lp

flexural - outside Lp

shear

yield penetration

total displacement

S2

(Uv=1.0% ; n=0.2)

`

0

10

20

30

40

50

60

70

80

90

100

00.511.522.5

Lateral Load (KN)

Drift (%)

flexural - within Lp

flexural - outside Lp

shear

yield penetration

total displacement

S3

(Uv=1.0% ; n=0.4)

0

10

20

30

40

50

60

70

80

90

100

00.511.522.5

Lateral Load (KN)

Drift (%)

flexural - within Lp

flexural - outside Lp

shear

yield penetration

total displacement

S4

(Uv=0.56% ; n=0.4)

Fig ure 11: Flexure, yield penetration, shear and total drift for column specimens S1-S4.

142

Australian Journal of Structural Engineering

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“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

measured curvature distribution as shown in gure

9 and summarised in table 5. Numerous empirical

models (Mattock, 1967; Priestley etal, 1996; Sawyer,

1964; Park & Paulay 1975) have been developed to

predict the equivalent plastic hinge length of ductile

columns using the general equation as a function

of shear span length (L), column width (D), and

diameter of main rebars (d

b

), as follows:

L

p

=

L +

D +

f

y

d

b

(15)

where L = shear span length, D = effective depth

of the section, and d

b

= diameter of longitudinal

reinforcement.

Specimen S3 with the highest axial load ratio

(n =0.4) and greatest longitudinal reinforcement

ratio (

v

=1.0%) had the smallest curvature and the

largest plastic hinge length. Further, the longitudinal

steel ratio had a greater effect on spreading the plastic

hinge region compared with the axial force ratio

(compare S1 and S2 when n = 0.2 with S3 and S4

when n = 0.4). The increase of axial load ratio from

n = 0.2 to n = 0.4 increased the plastic hinge length in

the order of 10%, while the increase of longitudinal

reinforcement ratio from

v

= 0.56% to

v

= 1.0%

increased the plastic hinge length by about 30%.

It was observed from the experimental tests that

the concrete spalling was localised and was limited

to the region where the rebar buckling occurred

which was approximately equal to the tie spacing.

Therefore, the spacing of stirrups can be considered

an upper limit of plastic hinge length for lightly

reinforced concrete columns. In contrast, the Park

& Paulay (1975) model provides a conservative

estimate for the lower bound limit of the plastic

hinge length as shown in table 5.

In this study, a computer program based on a

fibre section analysis (Park & Paulay, 1975) has

been developed to calculate moment-curvature

relationship of the column, which was then used

to predict the exural de ection by idealising the

curvature distribution into elastic and inelastic

regions as follows:

=

fe

=

(16)

The elastic displacement can be calculated by

integrating the curvature over the column height

as follows:

2

0

3

L

yy

fe

ML

xdx

EI

(17)

The additional inelastic rotation due to plastic

hinge formation at the base of the column can be

expressedas:

p

= (

u

–

y

)L

p

(18)

Hence, the lateral inelastic de ection at the top of the

column is given by:

22

pp

fi p u y p

LL

LLL

(19)

The analytical results using the experimental plastic

hinge length to estimate the exural displacement

showed good agreement with the experimental data

as presented in gure 12.

Table 4: Drift components for specimens S1-S4.

Specimen

Drift (%)

Max

drift

Total (measured)

drift

Flexure

within L

p

Flexure

outside L

p

Total

exure

Shear

Yield

penetration

S1 5.0 3.0 2.1 0.3 2.40 0.10 0.50

S2 2.5 2.0 1.2 0.4 1.60 0.10 0.30

S3 1.5 1.5 0.7 0.5 1.20 0.08 0.20

S4 1.5 1.5 1.0 0.3 1.30 0.06 0.17

Note: The total measured drift consists of the component drifts for which the measurements were reliable.

Table 5: Predicted and measured plastic hinge lengths for specimens S1-S4.

Model

αβ ξ

L

p

[mm]

S1 S2 S3 S4

Mattock, 1967 0.050 0.50 0.000 210 210 210 210

Priestley et al, 1996 0.080 0.00 0.022 238 285 285 238

Sawyer, 1964 0.075 0.25 0.000 165 165 165 165

Park & Paulay, 1975 0.000 0.50 0.000 132 133 133 132

Average plastic hinge length (experimental) 170 220 240 190

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5.2 Yield penetration displacement

Yield penetration behaviour has been widely

investigated over the last two decades (Zhao &

Sritharan, 2007). A classical moment-curvature

relationship has commonly been used to calculate

the yield penetration gap opening. The main problem

with this approach is that the steel stress and strain

obtained at the gap opening is based on strain

compatibility between the steel and the concrete,

whereas the presence of a gap opening and yield

penetration at the column base prevents the use of a

closed form solution to predict the neutral axis depth

because of strain incompatibility between the steel

and concrete at the connection interface. A trial-and-

error procedure is usually needed to evaluate the

neutral axis depth through a convergence process

of slip displacement, steel stress-strain, and yield

penetration length. However, this approach is quite

indirect and cumbersome and a closed form model to

predict the yield penetration displacement has been

developed and is presented in this section.

The proposed model and algorithm uses a simpli ed

stress-strain relationship (refer gure 13) together

with geometrical compatibility principles to solve

the strain incompatibility between the steel and

concrete (refer gure 14). A closed form algorithm

has been developed by modifying the bre cross-

section analysis to operate under displacement

control, such that the curvature is directly linked to

the displacement at the top of the column as follows:

1. Fix the increment of global displacement

top

, and

calculate the related rotation

slip

=

=

top

/L

(20)

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

Lateral Load (kN)

Flexural Drift (%)

Experiment Analysis

S1 (Uv=0.56% ; n=0.2)

0

10

20

30

40

50

60

70

80

90

100

0 0.5 1 1.5 2 2.5

Lateral Load (kN)

Flexural Drift (%)

Experiment Analysis

S2 (Uv=1.0% ; n=0.2)

`

0

10

20

30

40

50

60

70

80

90

0 0.5 1 1.5 2

Lateral Load (

kN

)

Flexural Drift (%)

Experiment Analysis

S3 (Uv=1.0% ; n=0.4)

0

10

20

30

40

50

60

70

80

90

00.511.52

Lateral Load (kN)

Flexural Drift (%)

Experiment Analysis

S4

(Uv=0.56% ; n=0.4)

Fig ure 12: Comparison of experimental and analytical exural drift components for column

specimens S1-S4. Note, drifts are plotted up to the point where the LVDTs were

removed during the experiment.

H

su

H

y

H

s

K=

ysu

ysu

ff

HH

f

su

f

s

f

y

K

1

Fig ure 13: Simplified stress-strain relationships

for yield penetration analysis.

L

L’

y

c

U

(

U

- y)

T

T

s

li

p

'

slip

'

to

p

G

A

B

O

L

d

Fig ure 14: Gap opening mechanism for yield

penetration analysis.

2. Estimate an initial value of concrete strain

c

,

and calculate the neutral axis depth (c) from

geometrical compatibility

• Evaluate curvature, for each case

Strain compatibility

= 3

top

/L

2

(21)

Strain compatibility

=

top

/L

(22)

• Evaluate neutral axis depth c =

c

=

c

/

(23)

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• Evaluate steel deformation at gap slip interface

slip

=

slip

(d – c) (24)

• Evaluate tensile steel strain

s

8

for

slip c

sinitsy

sb

f'

Ed

(25)

2

4

1

2

c

s

s slip y init

b

f'

E

d

for

s

<

y

(26)

where

init

= P/BDE

c

= initial strain due to

external axial load and member self-weight.

Fig ure 16: Measured and predicted yield penetration drift for column specimens S1-S4.

'

fl

'

yp

M

Fig ure 15: Moment-deflection relationship for

the yield penetration displacement

calculations for column specimen S1.

The related stress f

s

can then be obtained from

the steel stress-strain relationship.

• Evaluate the section equilibrium, so that the

net axial force (P) results,

C

c

– T

s

+ C

s

= P (27)

where C

c

is determined using moment-

curvature analysis. T

s

and C

s

can be obtained

from steel stress-strain relationship.

3. Iterate Step 2 with a new value of

c

until the

equilibrium equation is satisfied, and then

calculate the moment capacity with the updated

value of c and

c

222

c

sts scs

Cc D D

MTdCdP W

(28)

4. Calculate lateral load

F

h

= M/L (29)

5. Repeat the whole process for the next increment

of de ection

top

.

The solution of this algorithm provides the lateral

load-displacement relationship for the yield

penetration mechanism as shown schematically in

gure 15 for specimen S1. Good agreement between

the experimental and analytical results for the lateral

load versus yield penetration drift is demonstrated

in gure 16 for the four specimens.

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Australian Journal of Structural Engineering Vol 15 No 2

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5.3 Shear displacement

The displacement contribution from shear

deformation to the total deflection of a limited

ductile column is relatively small and hence the use

of a complicated shear deformation model is not

warranted. Further, since damage to stirrups was

barely found on any specimen post column failure,

the shear deformation model proposed by Park &

Paulay (1975) can be used. The total shear de ection

sh

can be calculated from the summation of the

elastic shear de ection in the uncracked portion

vuncrack

and the shear de ection in the cracked portion

vcrack

as follows:

sh

=

vuncrack

+

vcrack

(30)

A plot of predicted and measured shear de ections

for all specimens is shown in gure 17. The model

predicted shear deformations with an axial load ratio

of n = 0.4 (specimens S3-S4) is in good agreement with

experimental results and reasonable agreement for

specimens with an axial load ratio of n = 0.2 (S1-S2).

Overall, since the shear displacements are very small

for exure dominant slender columns (with shear

drifts up to 0.1% for the four specimens or less than

5% of the total drift) as shown in gure 18 and table

4, the model and predicted results are considered

reasonable and satisfactory.

5.4 Total lateral displacement

The total displacement consisting of the exural, yield

penetration and shear displacement components is

Fig ure 17: Measured and predicted shear drifts for column specimens S1-S4.

0

1

2

3

4

5

6

00.511.522.533.5

Drift (%)

flexural - within Lp

flexural - outside Lp

shear

yield penetration

total displacement

S1

(

U

v=0.56% ; n=0.2)

Fig ure 18: Shear displacement distribution and drift contributions for column specimen S1.

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Australian Journal of Structural Engineering

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“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

summarised in table 6. The proposed model was used

to construct the backbone force-drift curve which

showed good agreement with the experimental

hysteresis curves as shown in gure 19. Further,

tables 7 and 8 shows that the proposed model for

estimating the drift at both lateral load failure and

axial load failure showed good agreement with the

experimental results and is an improvement on the

existing models listed (refer to Wibowo et al (2014)

for further details).

6 SIMPLIFIED CODE BI-LINEAR MODEL

The simpli ed code bi-linear model is based on the

assumption underlying most force based seismic

codes of practice, where the inelastic behaviour is

represented by a ductility factor and over-strength

factor (ratio of maximum strength to factored

ultimate design strength). The ductility factor (ratio

of ultimate displacement to yield displacement) is

dependent on the level of ductility, and for lightly

Fig ure 19: Experimental hysteretic curves versus the backbone theoretical model curve for column

specimens S1-S4.

Table 6: Total drift for specimens S1-S4.

Results Parameters S1 S2 S3 S4 Stage

Theoretical prediction

Displacement

top

[mm]

1.90 1.70 2.20 2.20 Crack

12.60 11.90 11.60 11.30 Yield

19.80 20.50 12.50 12.30 Ultimate

Drift

top

[%]

0.16 0.15 0.19 0.19 Crack

0.85 0.89 0.97 0.94 Yield

1.65 1.71 1.04 1.03 Ultimate

Experimental Drift

top

[%]

0.13 0.13 0.20 0.20 Crack

0.75 0.75 1.00 1.0 Yield

1.71 1.73 1.12 1.01 Ultimate

Table 7: Comparison between predicted and measured drift at lateral load failure for column specimens.

Drift [%]

Specimen

S1 S2 S3 S4

Elwood & Moehle, 2003 2.4 2.3 1.7 1.8

Zhu et al, 2007 2.2 2.2 1.6 1.6

Proposed model 2.8 1.9 1.5 1.6

Experimental results 3.2 2.1 1.5 1.5

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Australian Journal of Structural Engineering Vol 15 No 2

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The effective stiffness I

eff

can be conservatively

estimated using FEMA356 (FEMA, 2000) to account

for tensioning stiffness effects in RC structures as

follows: I

eff

= 0.7I

g

for axial load ratio n 0.5, or

I

eff

=0.5I

g

for axial load ratio n 0.3. For 0.3 n < 0.5,

the value of I

eff

should be interpolated.

The maximum displacement

m

is calculated by

multiplying the yield displacement

yu

by the

ductility and over-strength value

as shown in

gure 20. In the Australian context, this results in

m

= 2.6

yu

(

=2.0×1.3 = 2.6), for limited ductile

columns with standard detailing.

The simplified bi-linear model is presented to

demonstrate the approach implicitly assumed in

force-based seismic codes of practice and to provide a

quick and conservative estimate of the displacement

at peak lateral load that can be used for initial seismic

performance checking using displacement principles.

The bi-linear model is not intended to predict the

drift at lateral load failure or axial load failure, but

provides a quick displacement checking method to

ascertain whether a more detailed study is needed.

The simplified code bi-linear model provides a

conservative estimate of the drift at peak lateral

load for each of the test results as shown in gure 21

(where it has been assumed that

= 0.8 and =1.3,

hence

= 1.04). Clearly for the experimental results

presented, each of the columns had significant

additional drift capacity beyond the maximum

displacement capacity (

m

=

yu

) implied in the

force-based code design approach.

7 CONCLUSIONS

Experimental research on four lightly reinforced

concrete columns has been undertaken with axial

Displacement

'

yu

'

m

=

P

'

yu

I

V

u

I

V

u

Base Shea

r

A

B

I

eff

'

yu

Fig ure 20: Simpli ed code bi-linear lateral

load-drift model.

Fig ure 21: Experimental load-drift curve versus the simpli ed code bi-linear model for column

specimens S1-S4.

Table 8: Column drift limits for specimens S1-S4.

Guidelines and models S1 S2 S3 S4

FEMA, 1997 1.1% 1.2% 0.9% 0.8%

FEMA, 2000 0.9% 1.1% 0.8% 0.7%

Elwood & Moehle, 2003 2.0% 2.0% 1.2% 1.0%

Ousalem et al, 2004 1.7% 1.7% 1.3% 1.2%

Zhu et al, 2007 1.7% 1.7% 1.3% 1.2%

Proposed model 4.8% 2.6% 1.8% 1.3%

Experimental 5.0% 2.5% 1.5% 1.5%

reinforced columns a low ductility factor of

= 2

and an over-strength factor = 1.3 is assumed in

many seismic codes of practice, such as AS1170.4

(Standards Australia, 2007).

The bi-linear model is constructed by estimating the

yield displacement

yu

from the factored ultimate

strength

V

u

divided by the effective stiffness I

eff

.

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Australian Journal of Structural Engineering

Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

load ratios n = 0.2-0.4, longitudinal reinforcement

ratio

v

= 0.56% to 1.0% and a very small transverse

reinforcement area ratio of

h

= 0.07% which was

signi cantly less than minimum code requirement.

The drift at axial load failure for all specimens

was at least 1.5% despite the poor detailing and

signi cantly greater than that predicted by current

design guidelines. The effect of the design parameters

can be summarised as follows:

• Increasing the axial load ratio reduces the

curvature ductility capacity as a result of an

increase of yield curvature and a decrease of

ultimate curvature.

• Increasing the axial load ratio from n = 0.2 to n =0.4

resulted in a reduction of the axial load failure drift

with the larger reduction found on columns with

a lower longitudinal reinforcement ratio.

• Increasing the longitudinal reinforcement ratio

decreased the drift at axial load failure for

specimens S1 and S2 with a lower axial load

ratio of n = 0.2, while the drift was reduced and

similar for both specimens S3 and S4 with an axial

load ratio n = 0.4 close to the balance point of the

interaction diagram.

The flexure, yield penetration and shear drift

components and their effect on the overall behaviour

have been presented in the paper. The flexural

components were the most dominant behaviour

for all specimens at around 75% to 85% of the total

drift due to the large aspect ratio of a = 4. The yield

penetration components for specimens S1, S2, S3

and S4 were quite small and around 10-20% of the

total drift, while the shear component was the least

dominating behaviour observed on all specimens at

around 2% to 5% of total drift.

The experimental tests clearly demonstrates the

reduced drift capacity of columns that are heavily

loaded, particularly as the axial load ratio approaches

or exceeds the balance point on the interaction

diagram. Such columns are considered highly

stressed under gravity loading and hence have

signi cantly reduced drift capacity. This observation

has significant implication for the earthquake

performance of structures that have heavily loaded

columns, since such structures will inherently have

reduced drift capacity under extreme lateral loading.

Designers can readily improve the earthquake

performance of their structures by ensuring the

columns are designed to remain in the tension

controlled region of the interaction diagram.

A theoretical model for predicting the exural, yield

penetration and shear displacements of a laterally

loaded column was developed and showed very good

correlation with the experimental results. Finally, a

simpli ed code bi-linear model was presented that

provides a quick and conservative estimate of the

displacement at peak lateral load for columns that

can be used for initial seismic performance checking

using displacement principles.

ACKNOWLEDGEMENTS

The nancial support from the ARC Discovery Grants

titled “Collapse modelling of soft-storey buildings”

(DP0772088) and “Displacement controlled behaviour

of non-ductile structural walls in regions of lower

seismicity” (DP1096753) is gratefully acknowledged.

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Vol 15 No 2

“Drift capacity of lightly reinforced concrete columns” – Wibowo, Wilson, Lam & Gad

ARI WIBOWO

Ari Wibowo is a Lecturer in Civil Engineering at University of Brawijaya in

Indonesia. He has a Bachelor of Engineering (Civil) from Sepuluh Nopember

Institute of Technology, Indonesia, Masters of Engineering from Bandung

Institute of Technology, Indonesia, and a PhD from Swinburne University of

Technology, Australia. His research interests include earthquake engineering,

structural dynamic, modelling and programming.

JOHN WILSON

John Wilson is Dean of Engineering and Industrial Sciences at Swinburne

University of Technology in Melbourne. Prior to joining Swinburne in 2005, he

was an academic at the University of Melbourne for some 14 years, and as a

consulting engineer for over 10 years with the SECV and Arups in their London

and Melbourne of ces. He has a Bachelor of Engineering degree from Monash

University, a Master of Science degree from University of California (Berkeley),

USA, and a PhD from University of Melbourne. He has a research interest and

expertise in structural systems, earthquake engineering, structural dynamics and

sustainable structures, and has consulted widely in these elds. Three Chapman

Medals and the Warren Medal awarded by Engineers Australia and the Gupta

Award by the ISET Journal of Earthquake Technology are acknowledgements

of his contributions to the research and practice of earthquake engineering.

NELSON LAM

Nelson Lam, Reader in Civil Engineering at The University of Melbourne, is

an internationally recognised expert in earthquake engineering and structural

dynamics. He has 30 years of experience in the professional practice and

research in structural engineering. In the past 20 years, he has been working

in the specialised eld of earthquake engineering and impact dynamics. His

achievement in research and knowledge transfer in this eld was recognised

by the award of the Chapman Medal (1999, 2010) and Warren Medal (2006) by

Engineers Australia, and the Best Paper Award (2004-2007) by the ISET Journal

of Earthquake Technology. He served as member of the committee for developing

the current edition of the Standard for Earthquake Actions in Australia AS1170.4

(2007), and was co-author and co-editor of the Standards Commentary. His early

career as structural engineer was with Scott Wilson International throughout

the 1980s and attained British chartered engineer status during that period.

EMAD GAD

Emad Gad is a Professor in Civil Engineering at Swinburne University of

Technology in Melbourne. Prior to this appointment he worked as a senior

academic at Melbourne University and research scientist at CSIRO. He has

a Bachelor of Engineering (Civil) from Monash University and a PhD from

Melbourne University. Emad has research interest and expertise in structural

dynamics and modelling. In addition to teaching and research he has also been

involved in specialist consulting. He is a Fellow of Engineers Australia and past

Chairman of the Structural Branch, Victoria Division.