Effect of slab bottom reinforcement on seismic performance...

Magazine of Concrete Research, 2012, 64(4), 317–334

http://dx.doi.org/10.1680/macr.11.00012

Paper 1100012

Received 19/01/2011; last revised 15/03/2011; accepted 25/03/2011

Thomas Telford Ltd & 2012

Magazine of Concrete ResearchVolume 64 Issue 4

Effect of slab bottom reinforcement onseismic performance of post-tensioned flatplate framesHan, Moon and Park

Effect of slab bottomreinforcement on seismicperformance of post-tensionedflat plate framesSang Whan HanProfessor of Architectural Engineering, Hanyang University, Seoul, SouthKorea

Ki Hoon MoonPost Doctoral, Department of Architectural Engineering, HanyangUniversity, Seoul, South Korea

Young-Mi ParkChief Engineer, Doosan Engineering and Construction Company, SouthKorea

The purpose of this study is to evaluate the seismic performance of gravity-designed post-tensioned flat plate frames

with and without slab bottom reinforcement passing through the column. In low and moderate seismic regions, the

seismic demands may not control the design, and buildings are often designed considering only gravity loads. This

study focuses on the seismic performance of gravity load designed post-tensioned flat plate frames. For this purpose,

three-, six- and nine-storey post-tensioned flat plate frames are designed considering only gravity loads. For

reinforced concrete flat plate frames, continuous slab bottom reinforcement (integrity reinforcement) passing

through the column should be placed to prevent progressive collapse; however, for the post-tensioned flat plate

frames, the slab bottom reinforcement is often omitted since the requirement for the slab bottom reinforcement for

post-tensioned flat plates is not clearly specified in ACI 318-05. This study evaluates the seismic performance of the

model frames by conducting non-linear static pushover analyses and non-linear response history analyses. For

conducting non-linear response history analyses, six sets of ground motions are used as input ground motions, which

represent two different hazard levels (return periods of 475 and 2475 years) and three different locations (Boston,

Seattle and Los Angeles). An analytical model is developed for emulating the non-linear hysteretic behaviour and the

failure mechanism of post-tensioned slab–column connections. This study shows that gravity-designed post-

tensioned flat plate frames have some seismic resistance. In addition, the seismic performance of post-tensioned flat

plate frames is significantly improved by placing slab bottom reinforcement to pass through the column.

IntroductionIn recent years post-tensioned (PT) flat plate frames have become

increasingly popular. This system is very effective since pre-

stressed concrete leads to a reduction in construction period,

greater span-to-depth ratio, and improved crack control compared

with conventional reinforced concrete (RC) flat plate frames. In

low and moderate seismic regions, most low- to mid-rise frames

have been designed only considering gravity loads (Bracci et al.,

1995; Han et al., 2004). However, satisfactory seismic perform-

ance of gravity-designed flat plate frames may not be guaranteed

when the frames are subjected to the ground motions for design

level and maximum considered earthquakes specified in current

seismic design provisions.

According to ACI 318-05 (ACI Committee 318, 2005), chapter 18,

minimum top bonded reinforcement is required for PT gravity flat

plate systems; however, for these PT gravity systems, minimum

bottom reinforcement is usually not required in the connection

regions. For RC flat plate systems, ACI 318-05 (ACI Committee

318, 2005), section 7.13.25 requires at least two continuous bottom

reinforcing bars (for interior connections) or two bottom bars with

908 standard hooks (for exterior connections) passing through the

joint core as an integrity steel requirement. ACI 352.1R-89 (ACI–

ASCE Joint Committee 352, 1997) supplements the ACI 318

integrity steel requirement in greater detail. It is noted that some

engineers argue that the integrity steel requirement may also be

applied to PT flat plate systems, since the ACI 318 (ACI

Committee 318, 2005) and 352 (ACI–ASCE Joint Committee 352,

1997) provisions do not clearly state that no bottom integrity

reinforcement is required for PT slabs. Typically, it is believed that

draped tendons preclude catastrophic collapse of PT slabs after

punching failure, so that bottom bonded reinforcement is often not

provided in practice; however, the bottom reinforcement may be

needed, not only for the purpose of collapse prevention, but also to

resist sagging (positive) moments which may occur for the gravity

systems subjected to inelastic deformations.

317

However, for PT flat plate frames, slab bottom reinforcement

passing through the column has often been omitted, for several

reasons, as follows:

(a) prior experimental observation revealed that continuously

draped tendons passing through the column core are effective

in avoiding the progressive collapse of PT slab–column

connections (Mitchell and Cook, 1984)

(b) the minimum slab reinforcement requirement specified for

RC flat plates frames can be relieved in PT flat plate frames

if the tendons provide at least 0.7 MPa of average

compressive stress ( fpc) on the gross concrete section, and the

maximum spacing of tendons does not exceed the spacing

limit specified in ACI 318-05 (ACI Committee 318, 2005),

section 7.12.3

(c) current building codes do not clearly specify the requirement

of bottom reinforcement for PT flat plates.

Previous studies (Han et al., 2006a, 2006b, 2009; Martinez-

Cruzado, 1993; Qaisrani, 1993) reported that a slab sagging

moment could occur at PT slab–column connections under lateral

force associated with design earthquake specified in current

seismic design provisions; therefore, sufficient slab bottom rein-

forcement must be placed to avoid the formation of large cracks.

Han et al. (2009) demonstrated that slab bottom reinforcement

passing through the column placed in accordance with ACI–

ASCE 352 R.1 (ACI–ASCE Joint Committee 352, 1997) is

effective in resisting unexpected sagging moments at PT slab–

column connections. However, no study has been conducted to

evaluate the effect of slab bottom reinforcement passing through

the column on the seismic performance of PT flat plate frames.

This study investigates the seismic performance of gravity-

designed PT flat plate frames with and without slab bottom

reinforcement passing through the column. For this purpose,

three-, six- and nine-storey PT flat plate frames are designed

considering only gravity loads. An analytical model is developed

for emulating the hysteretic behaviour of PT slab–column

connections, which is able accurately to predict connection failure

without the numerical convergence problem associated with inter-

action among non-linear springs at PT slab–column connections.

For evaluating the seismic performance of PT flat plate frames,

non-linear static pushover analyses and non-linear response

history analyses (RHAs) are conducted. SAC 10/50 and 2/50

ground motions for Boston, Seattle and Los Angeles (LA) are

used as input ground motions for non-linear RHA, where 10/50

and 2/50 represent 10% and 2% exceedence probability in 50

years, which corresponds to return periods of 475 and 2475 years,

respectively.

Analytical model for PT flat plate framesAn analytical model for PT slab–column connections is proposed

which is able accurately to depict connection failures without the

numerical convergence problem caused by interaction among the

non-linear springs installed for simulating the cyclic behaviour of

PT slab–column connections. Each non-linear spring will simulate

the punching shear failure mechanism; thus, only two non-linear

springs are necessary for representing the hysteretic behaviour of

interior slab–column connections. Since only two independent

springs are used in the proposed connection model, the numerical

convergence problem does not occur. Also the proposed connec-

tion model can predict three different connection failure mechan-

isms, which is more realistic than using one failure mechanism

defined by rotation drift limit (Banchik, 1987). Figure 1 shows

constituent elements for the connections, columns and slabs in PT

flat plate frames. A detailed description for each model is provided

below.

Slab model

As shown in Figure 1, slabs were modelled using an elastic beam

element, which has an effective beam width (ÆL2) and slab depth,

where Æ is the effective beam width coefficient and L2 is the slab

width. Effective beam width coefficients, Æi and Æe, for interior

and exterior frames respectively, were calculated using Equation

1 proposed by Banchik (1987)

Æi ¼ 5c1

L2

þ 1

4

L1

L2

� �1

1� �2

Æe ¼ 3c1

L2

þ 1

8

L1

L2

� �1

1� �21:

where c1 is the column dimension in the loading direction, � is

Poisson ratio of concrete, and L1 and L2 are the slab length and

width, respectively. To consider the effect of cracking, a stiffness

reduction factor (�) of 0.33 is used. Qaisrani (1993) reported that

� of 0.33 is also valid for PT flat plates.

Column model

A column section was grouped in several patches such as a cover,

and core concrete, and steel layers using a fibre section model

(Figure 2(a)) embedded in software OpenSees (2007). The fibre

section model has been demonstrated to be an appropriate model

to simulate the non-linear flexural behaviour of beams and

columns (Spacone et al., 1996). For a numerical model, the

plastic behaviour of steel is represented by a bilinear model

(Figure 2(b)). The stress and strain relation for the concrete

(Figure 2(c)) is modelled in accordance with Hognestad et al.

(1995).

Connection model

The hysteretic behaviour of PT slab–column connections are

complex since there are various failure mechanisms as a result of

the interaction between flexure and punching shear, as well as the

contribution of tendons and reinforcing bars.

In this study, a pinching model embedded in OpenSees (2007) is

used to simulate the hysteretic behaviour of slab–column connec-

tions shown in Figure 3. Coefficients for the pinching model are

318


Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park

calibrated to simulate accurately the actual hysteretic behaviour

obtained from previous experiments (Han et al., 2006a). Pinching

coefficients of 0.5 and 0.2 are used for the deformation pinching

coefficient (px) and load pinching coefficient under the reloading

process (py), respectively. The coefficient of displacement ducti-

lity (�) is set to 0.5, which represents the stiffness in unloading

branches.

To simulate failure mechanisms, this study assumes that PT flat

plate connections fail by one of two different failure modes:

shear-dominated failure mode or flexure-dominated failure mode.

In the shear-dominated failure mode, a slab–column connection

fails by punching shear before the slab reinforcement yields,

whereas in flexure-dominated failure mode, slab reinforcement

yields prior to punching shear failure occurring at the connection.

Resultant concentrated loaddue to gravity

Non-linearslab connection spring(zero length element)

Fibre section

Fixed boundary condition

L2 α�L2

Effective beam width model(elastic beam element)

Figure 1. Planar frame model

Y

Z

Core concrete

Cover concrete

Reinforcing steel bar

(a)(b) (c)

fy

�fy

Stre

ss: M

Pa

0·1Es

Es�εy

�εy

Strain

f �c Confined concrete

0·8f �c

ε0 εu,c

Unconfinedconcrete

Figure 2. Fibre section model for columns: (a) fibre model;

(b) rebar material; (c) concrete material

319



Shear dominated failure mode

Figure 4(a) depicts the shear-dominated failure mode. When the

shear stress due to an unbalanced moment at the connection

exceeds the punching shear capacity before the slab reinforce-

ment within the effective slab width (c2 + 3h) yields, a sudden

brittle punching shear failure takes place, where c2 is the column

width and h is the slab thickness. Such a failure often occurs

under a high gravity shear ratio (GSR ¼ V g=�Vc), where Vc is

the punching shear strength at the slab–column connection, Vg is

the direct shear force due to vertical gravity loads acting on the

slab, and � is the strength reduction factor. The unbalanced

moment (Munb) at the connection corresponding to the punching

shear capacity can be estimated using Equation 2, which is a

rearrangement of the eccentric shear model specified in ACI 318-

05.R11.11.7.2 (ACI Committee 318, 2005).

Munb ¼ vn �V g

bod

� �Jc

cªv2:

where vn is the nominal shear stress, bo is the perimeter of slab

shear critical section, Jc is the property of assumed critical

section analogous to a polar moment of inertia, c is the distance

between the centroid and edge of the shear critical section, and

ª� is the fraction of unbalanced moment transferred by the

eccentricity of the shear at the slab–column connection, calcu-

lated using Equation 3

ª� ¼1

1þ (2=3)ffiffiffiffiffiffiffiffiffiffiffiffib1=b2

p3:

where b1 is the dimension of the critical section in the loading

direction and b2 is the dimension of the critical section perpendi-

cular to b1: Note that the unbalanced moment based on the

eccentric shear stress model does not include the effect of the

factored loading or the strength reduction factor because the

expected behaviour of the connection is to be evaluated.

The nominal shear stress (vn) in Equation 2 for the PT slab–

column connections can be determined using Equation 4

vc ¼ �p

ffiffiffiffiffif 9c

pþ 0:3 f pc

� �þ Vp

bod4:

where �p is the smaller of 0.29 and 0.083 (Æsd/bo + 1.5), Æs is 40

for interior columns, 30 for edge columns, and Vp is the vertical

component of all effective prestressed forces crossing the critical

section.

Flexure-dominated failure mode

Slab–column connections fail in a ductile manner when the Munb

value that corresponds to the punching shear capacity (Equation

2), is greater than the Munb value that results in slab reinforce-

ment yielding (see Figures 4(b) and 4(c)). The unbalanced

moment Munb that causes slab yielding can be calculated by the

following equations

Munb ¼ M�y þ M�y for interior connections5a:

Munb ¼ M�y for exterior connections5b:

where Mþy and M�y are sagging and hogging (negative) slab yield

moments, respectively, calculated by using Equations 6a and 6b.

Figure 5(a) shows the sagging and hogging slab yield moments.

M�y ¼ (Asp fpe þ Ast f y) jd � M g6a:

Force ( )F

Force ( )F

smax

eh

Ep sy max

emax

ech

Displacement ( )Δ

Displacement ( )Δ

e e p e ee e p s E

e e p e e

1 1 y max u

2 max y max

ch 1 x 2 1

( )(1 ) /

( )

� � �

� � �

� � �

(a)

( , )e smax max

Ee

Eeμ��

μ /� e emax 1p

(b)

Figure 3. Hysteretic model for PT slab–column connections

(OpenSees, 2007): (a) pinching parameter; (b) unloading

parameter

320



Mþy ¼ (Asb fy) jd þ M g6b:

where M g is the slab moment due to gravity loads, fpe is the

effective stress due to the post tensioning of tendons considering

stress relaxation, Asp and Ast are the area of strands and top

reinforcement located within the slab–column strip, respectively,

Asb is the area of bottom reinforcement passing through the

column, fy is the nominal yield strength of reinforcement, d is

the effective depth of slab, and jd is the distance between

resultant forces of tension and compression, which is calculated

using Equations 7 and 8.

j ¼ d � kd

37:

k ¼ (rþ r9)n2 þ 2(rþ r9d9

d)n

� �1=2

� (rþ r9)n8:

where r (r9) is the ratio of the area of tension (compression)

reinforcement to the slab effective cross-sectional area [¼ Asb=ld

(¼ (Ast þ Asp)=(ld))], l is the width of the column strip of the

slab, d9 is the distance from the extreme compression fibre to the

Axial force

Rigidjoint

Munb

0·1Munb 0·1Mu0·1Mn

θ θ θ

θ

(a)Shear failure mode

M M M

Mn MnMy

0·02K� Kpost-yield

θn

(b)Flexural shear failure

Mode 1�

θu

θu

θn

θn

(c)Flexural shear failure

Mode 2�

MMnMuMy

Connection model

Connectionspring

A

A

V

be

hElastic beam element

Section A–A

ColumnY

ZCore concrete

Core concrete

Cover concrete

Cover concrete

Reinforcing steel bar

(a)(b) (c)

fy

�fy

Stre

ss: M

Pa

0·1Es

Es�εy

εy

Strain

f �c

0·8f �c

ε0 εu,c

Figure 4. Analytical model at PT slab–column connections:

(a) fibre model; (b) rebar material; (c) concrete material

321



centroid of the bottom reinforcement, and n is the ratio of the

elastic modulus of steel to the concrete (¼ Es=Ec).

Martinez-Cruzado (1993) and Qaisrani (1993) reported that post-

yielding lateral stiffness (K9) of PT flat plate connections ranges

from 1 to 3% of the elastic lateral stiffness. In this study, 2% of

the elastic stiffness was used as the post-yielding stiffness for the

PT slab–column connections. In the flexure-dominated failure

mode, the connection failure occurs when reaching unbalanced

moment becomes slab flexural strength, punching shear strength,

or connection rotation limits for punching shear failure. The

unbalanced moment Munb at the connection corresponding to the

punching shear capacity is calculated by using Equation 2. The

unbalanced moment Munb, corresponding to slab flexural

strength, is calculated using Equations 5a and 5b with a slight

modification in which slab yield moments (Mþy , M�y ) are re-

placed by slab nominal flexure strength (Mþn , M�n ). Sagging and

hogging slab nominal flexural strength, Mþn and M�n can be

calculated using Equations 9a and 9b, respectively. Figure 5(b)

shows the sagging and hogging slab yield moments

Mþn ¼ (Asb fy) d � a

2

� �þ M g

9a:

M�n ¼ (Asp fps þ Ast f y) d � a

2

� �� M g

9b:

where fps is the stress in tendons at nominal flexural strength and

a is the depth of equivalent rectangular stress block of the slab in

the ultimate state that can be determined using Equation 10.

a ¼ Asp f ps þ Ast f y � Asb f y

0:85 f 9cb10:

Punching shear failure in slab–column connections occurs not

only by reaching punching shear capacity but also by reaching a

connection rotation limit. ACI 318-05 (21.13.6) (ACI Committee

318, 2005) specifies a drift ratio at punching shear failure for RC

flat plate connections. This study proposed an equation for

estimating drift ratio (Łu) corresponding to punching shear force

in PT slab–column connections. For this purpose, regression

analysis is conducted on test results for PT slab–column connec-

tions to establish an equation for calculating Łu with respect to

gravity shear ratio V g=�Vc: Equation 11 is the proposed equation

for Łu, and Figure 6 shows actual and predicted Łu obtained from

the test results and Equation 11.

Łu ¼ 0:06� 0:06V g

�Vc11:

Verification of the analytical model

PT slab–column connections

The hysteretic behaviour of the PT slab-column connections

reported by Han et al. (2006a, 2006b, 2009) is simulated by using

the numerical connection model developed by this study. Soft-

ware OpenSees (2007) is used. Test variables in the previous

study are:

(a) gravity shear ratio (�Vc=V g)

(b) tendon distribution (banded or distributive arrangement)

(c) connection locations (interior and exterior connection)

(d ) existence of bottom reinforcement passing through the

column core.

Three interior connection specimens (PI-D30, PI-D50, PI-D50X)

and one exterior specimen (PE-D50) are considered for verifying

the accuracy of the numerical connection model, where ‘D’

indicates the distributed tendon arrangement, 30 and 50 denote

the two levels of gravity loads in terms of �Vc=V g, and ‘X’

E W–

M�y M�

y

CfMg

Tf

jd jdFront Back

f �c

f �c

Tb

Mg

Cb

(a)

E W–

M�n M�

n

CfMg

Tf

Front Back

0·85 f �c

0·85 f �c

Tb

Mg

Cb

(b)

d �a

2d �

a

2

Figure 5. Analytical model at PT slab–column connections:

(a) slab yield flexural strength; (b) slab nominal flexural strength

322



stands for no slab bottom reinforcement passing through the

column. The detail information on these specimens is available in

Han et al. (2006a, 2006b, 2009).

Figure 7 shows the hysteretic behaviour of slab–column connec-

tion specimens PI-D30, PI-D50, PI-D50X and PE-D50 obtained

from the numerical analyses and experiments. From this figure, it

can be seen that the hysteretic curves obtained from the

numerical analyses agree well with those obtained from the

experiment. The occurrence of failure is also predicted by the

proposed analytical connection model with good precision.

Two-storey PT flat plate frame

The proposed analytical model for connections is used to

simulate the dynamic response of the PT flat plate frame tested

by Kang and Wallace (2005). Figure 8 shows the test specimen,

which has two storeys and two bays in both orthogonal directions.

Distributed tendons are arranged in the loading direction. A

ground motion (CHY087) recorded during M 7.6 Chi-Chi earth-

quake in Taiwan in 1999 is used as input ground motion for the

analyses as used for the shaking table test.

Figure 9 demonstrates that non-linear response history analysis

(RHA) using the proposed analytical model accurately predicts

the actual envelope curve for relating base shear to roof drift

ratio. Maximum base shear forces obtained from the experiment

and the numerical analysis are 204 kN and 210 kN, respectively.

Figure 10 shows the displacement history of the PT flat plate

frame obtained from the shaking table test and non-linear RHA

using the proposed analytical model. This figure shows that actual

displacement history is predicted by non-linear RHA using the

proposed analytical model with good precision. Table 1 shows

important response quantities obtained from the experiment and

analyses. The ratios of actual value to the predicted value for

base shear, roof drift ratios and fundamental periods are 1.03,

1.03 and 1.04, respectively.

Seismic performance evaluation

Model frames

This study considers three-, six- and nine-storey PT flat plate

frames as model frames, which are designed only for gravity

loads according to ACI 318-05 (ACI Committee 318, 2005). The

unit weight for office buildings is assumed to be 23.5 kN/m3: An

additional dead load (e.g. partition walls) of 0.5 kPa and a live

load of 2.0 kPa are accounted according to international building

code IBC-06 (International Code Council, 2006). Figure 11

shows the model frames. Concrete compressive strength, rein-

forcement yield strength and tendon tensile strength are assumed

to be 30, 400 and 1890 MPa, respectively. The number of

Drif

t ra

tio (t

otal

rot

atio

n) a

t pu

nchi

ng

0·09

0·08

0·07

0·06

0·05

0·04

0·03

0·02

0·01

0

0 0·1 0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·0

Gravity shear ratio, ( / ), where (3·5 0·3 )V V V f f b dg o o c1/2

pc o� � �

Cyclic load history (Kang and Wallace, 2005)

Monotonic load history (Kang and Wallace, 2005)

Monotonic load history (Han , 2006a)et al.

Monotonic load history (Han , 2006b)et al.

θu 0·06 0·06 (best fit)� �

Best fit � σRes

Best fit � σRes

ACI 318-05 21.11.5 Limit

σRes: Standard deviation of residuals aroundthe regression line

Vg

φVc

Figure 6. Drift ratio at punching failure plotted against gravity

shear ratio

323



post-tensioning tendons and the post-tensioning force per tendon

are determined such that approximately 100% of slab dead weight

is balanced. The resulting average compressive stress ( fpc) of

1.21 MPa is within the allowable range of 0.88–3.44 MPa as

specified in ACI 318-05 (ACI Committee 318, 2005). According

to section 18.9 of ACI 318-05, minimum bonded top reinforce-

ment ( fy ¼ 352 MPa) is placed within an effective slab width of

c2 + 3h. A square column cross section (30 cm 3 30 cm) is used

(a) (b)

(c) (d)

Analysis

Experiment

Roof drift ratio: %

�10 �8 �6 �4 �2 0 2 4 6 8 10 �8 �6 �4 �2 0 2 4 6 8 10

80

60

40

20

0

�20

�40

�60

40

30

20

10

0

�10

�20

�30

�40

Late

ral l

oad:

kN

Figure 7. Hysteretic curves of PT slab–column connections:

(a) PI-D30; (b) PI-D50; (c) PI-D50X; (d) PE-D50

Testingdirection

Testingdirection

E W

2845 28452845 2845

Unit: mm

305

953

991

254

1168

2642

1168

E W

S

N

Figure 8. Two-storey PT flat plate frame test specimen (Kang and

Wallace, 2005)

324



250

200

150

100

50

0

Base

she

ar: k

N

Roof drift ratio: rad0 0·01 0·02 0·03 0·04 0·05

ExperimentAnalysis

Figure 9. Envelope curves

Experiment

Analysis

9 11 13 15 17 19 21

0·05

0·03

0·01

�0·01

�0·03

Roof

drif

t ra

tio: r

ad

Time: sec

Figure 10. Response history of the test frame

Type Results Analysis

(1)

Experiment

(2)

(1)

(2)

Push-over Base shear: kN 209.87 203.91 1.03

Dynamic Roof drift ratio: rad 0.039 0.038 1.03

Fundamental period: s 0.56 0.54 1.04

Table 1. Comparison of response quantities obtained from non-

linear RHA and experiments

Thre

e ba

ys @

8 m

Thre

e ba

ys @

8 m

Thre

e ba

ys @

8 m

Three bays @ 8 m Three bays @ 8 m Three bays @ 8 m

(a)

(b)

Thre

e @

3·5

m

Six

@ 3

·5 m

Nin

e @

3·5

m

1·00

0·64

0·22

1·00

0·90

0·73

0·51

0·29

0·09

1·00

0·97

0·90

0·79

0·66

0·51

0·35

0·19

0·06

Figure 11. (a) Floor plans and (b) elevation of model frames

325



and slab thickness is assumed as 220 mm. Slab moments and

shear forces due to factored gravity loads are determined by

conducting elastic analysis using Midas/Gen (Version 6.3.2)

(Midas IT, 2004).

For PT flat plate frames having slab bottom reinforcement

passing through the column, the amount of slab bottom reinforce-

ment is calculated by the following equation, which is specified

in ACI–ASCE 352 (ACI–ASCE Joint Committee 352, 1997)

Asm ¼Æøu L1 L2

� f y12:

where Asm is the minimum area of continuous slab bottom rebar,

øu is the gravity loads, and Æ is the coefficient determined

according to connection location (Æ ¼ 1/3 for exterior, and

Æ ¼ 1/2 for interior)

The fundamental period of the frames is estimated using Midas/

Gen software (Midas IT, 2004), which is summarised in Table 2.

Figure 12 shows the reinforcement and tendon arrangements on

the fifth floor of the six-storey building.

Non-linear static pushover analysis

This study conducts non-linear static pushover analyses for the

PT flat plate frames using the software OpenSees (2007) with the

developed connection model. The vertical distribution of lateral

force used in the analyses is shown in Figure 11(b), which is

based on the first mode inertial force profile. From the analyses,

pushover curves are obtained, relating base shear coefficient

(V/W) to the roof drift ratio (Łroof ) or the maximum storey drift

ratio (Łmax). V is the base shear and W is the building weight. The

base shear is obtained by summing shear forces of the columns in

the first storey.

As shown in Figure 13, the effect of slab bottom reinforcement

on the behaviour of PT flat plate frames seems negligible within

the elastic range, whereas the effect is very significant beyond the

elastic range. Table 3 summarises the maximum base shear, and

maximum roof and storey drift ratios (Łmax�roof and Łmax�storey),

which is determined when lateral strength reduces to 80% of the

maximum lateral strength.

Maximum lateral strength of three-, six- and nine-storey PT flat

plate frames having slab bottom reinforcement is 189%, 189%

and 212% larger than those of the corresponding frames without

slab bottom reinforcement, respectively. Moreover, Łmax�roof and

Łmax�storey of three-, six- and nine-storey PT flat plate frames

having slab bottom reinforcement is 292%, 250% and 222%, and

373%, 235% and 240% larger than those of the corresponding

frames without slab bottom reinforcement, respectively. Thus,

maximum strength and maximum drift capacity of tested PT flat

plate frames are significantly increased by placing slab bottom

reinforcement passing through the column.

Figure 14 shows the vertical distribution of storey drift ratio with

respect to a given roof drift ratio. At a roof drift ration of 0.5%,

the distributions of storey drift ratios for PT flat plate frames with

and without slab bottom reinforcement are similar. With increas-

ing of the roof drift ratio, the location of maximum interstorey

drift ratio moves to the lower storeys, whereas, for the frames

without slab bottom reinforcement, the location does not consis-

tently move to the lower storeys.

In FEMA 273 (FEMA, 1997), limiting interstorey drift ratios for

life safety (LS), and collapse prevention (CP) for RC moment-

resisting frames are specified as 0.02 and 0.04, respectively.

When maximum interstorey drift ratio (Łmax) reaches 0.02 for LS,

the roof drift ratios of three-, six-, and nine-storey frames having

slab bottom reinforcement are 0.02 0.018, 0.015 and 0.013,

respectively, whereas the roof drift ratios of the frames without

slab bottom reinforcement are 0.016, 0.013 and 0.010, respec-

tively. Note that roof drift ratio is the lateral displacement at the

roof level normalised by building height. This indicates that the

PT flat plate frames without slab bottom reinforcement reaches

limiting interstorey drift ratio for LS at a smaller roof drift than

PT flat plate frames having slab bottom reinforcement. Similar

observation can be made for CP (Łmax ¼ 0.04). Furthermore, it is

observed that, when Łmax reaches limiting drift ratios for LS or

CP, the roof drift ratio becomes smaller as the number of storeys

increases irrespective of the existence of slab bottom reinforce-

ment.

Figure 15 shows locations of plastic hinges and punching shear

failures when Łmax reaches the limiting drift ratios for LS and CP.

At the limiting drift ratio for LS (¼ 0.02), plastic hinges are

detected in the frames having slab bottom reinforcement; how-

ever, punching shear failure does not occur, whereas plastic

hinges as well as punching shear failure were observed in the

frames without slab bottom reinforcement.

At the limiting drift ratio for CP (¼ 0.04), the PT flat plate

frames having slab bottom reinforcement also experience punch-

ing shear failure (Figure 15(b)). It is, however, noted that at Łmax

of 0.04 (CP), fewer members experienced punching failure in the

PT flat plate frames having slab bottom reinforcement than the

corresponding frames without slab bottom reinforcement; eight as

opposed to 18 for three-storey frames, 14 as opposed to 36 for

six-storey frames, and 21 as opposed to 38 for nine-storey frames.

Therefore, the frames without slab bottom reinforcement are

more susceptible to experiencing brittle punching shear failure

during earthquakes.

Structure Three-storey Six-storey Nine-storey

Fundamental period: s 1.00 1.94 2.99

Table 2. Fundamental periods of the model frames

326



Non-linear response history analyses

This study evaluates the seismic performance of three-, six- and

nine-storey PT flat plate frames with or without slab bottom

reinforcement passing through the column. For this purpose,

non-linear response history analyses are conducted using Open-

Sees (2007) with the developed connection model. Ground

motions for LA, Seattle and Boston are considered as input

ground motions, which are classified as seismic design cate-

gories (SDC) E, D and B according to IBC-06, as shown Figure

16. For each site, 20 10/50 and 20 2/50 SAC ground motions

are used, where 10/50 and 2/50 represent 10% and 2%

exceedence probability for 50 years, respectively. To calibrate

average response spectrum close to the design response spectrum

in FEMA 273 (FEMA, 1997), scaling factors are found accord-

ing to the procedure used by Yun et al. (2002), and are 0.9 and

0.83, 0.73 and 0.77, and 0.63 and 0.73 for LA, Seattle and

Boston 10/50 and 2/50 ground motions, respectively. The design

response spectrum (FEMA, 1997) is also shown in Figure 17. At

each site, the scaling factor is multiplied with each ground

motion. Figure 17 shows 20 scaled individual and average

122501160830

8000

2000

4000

2000

700

2600

700

: 12·7 mm seven-strand tendon

: HD13 bar

*Note

2300

1040

1040

D10

@30

0D

10@

300

With or without structural integrity bottom bar6-HD13@100(B)

6-HD13@100(B)

3-HD13@100(B)

HD13(T) HD13(T)

3-HD13@100(B)

12·7 mm seven-strand tendon

500 7500 500 3750

12250

220

Figure 12. Arrangement of reinforcement and tendons

327



0·35

0·30

0·25

0·20

0·15

0·10

0·05

0·30

0·25

0·20

0·15

0·10

0·05

0·25

0·20

0·15

0·10

0·05

00 0·01 0·02 0·03 0·04 0 0·01 0·02 0·03 0·04 0·05

(a)

(c)

(e)

(b)

(d)

(f)

With bottom bar With bottom barWithout bottombar

Without bottombar

V W/

V W/

V W/

Roof drift ratio: rad θmax: rad

Figure 13. Pushover curves for model frames: (a) three-storey;

(b) three-storey; (c) six-storey; (d) six-storey; (e) nine-storey;

(f) nine-storey

Storey Max. base shear: kN Łmax�roof Łmax�storey

Three-storey With bottom bar (1) 1147 0.035 0.041

Without bottom bar (2) 607 0.012 0.011

(1)=(2) 1.89 2.92 3.73

Six-storey With bottom bar (3) 1712 0.025 0.040


(3)=(4) 1.89 2.50 2.35

Nine-storey With bottom bar (5) 2303 0.020 0.036


(5)=(6) 2.12 2.22 2.40

Table 3. Summary of maximum base shear and maximum roof

drift ratio

328



acceleration response spectra for 5% damped single-degree-of-

freedom systems. In Figure 17, the fundamental period of three-,

six- and nine-storey frames is also marked.

Figures 18(a)–18(c) show median storey drift ratios under SAC

10/50 and 2/50 ground motions for Boston, Seattle, and LA.

Table 4 summarises the maximum storey drift ratio, Łmax, which

is the largest value among median storey drift ratios throughout

the building. According to FEMA 273 (FEMA, 1997), the basic

safety objective (BSO) is satisfied when assuring life safety and

With bottom barWithout bottom bar

3

2

1

6

5

4

3

2

1

9

8

7

6

5

4

3

2

1

0

Three-storey

Six-storey

Nine-storey

θroof 0·5%� 1·0% 1·6% 1·8% 3·2% 2·6% 3·1% 3·4%

θroof 0·5%�

θroof 0·5%�

1·0% 1·3%1·5% 2·3% 2·4% 2·5% 2·6%

1·0% 1·3% 1·7% 1·9% 2·1%

2·0%

Interstorey drift ratio: rad0 0·01 0·02 0·03 0·04

Figure 14. Distribution of storey drift ratio of three-, six- and

nine-storey PT flat plate frames

329



collapse prevention against 10/50 and 2/50 ground motions,

respectively.

Boston

As shown in Figure 18(a), the vertical distribution of median

storey drift ratios for the frames having slab bottom reinforce-

ment is similar to that for the corresponding frames without slab

bottom reinforcement. This can be attributed to the fact that

ground motions for Boston are small so that all frames behave in

the elastic range irrespective of the existence of slab bottom

reinforcement. Furthermore, for Boston 10/50 and 2/50 ground

motions, Łmax for all the frames does not exceed the limit drift

ratios for LS (¼ 0.02) as well as CP (¼ 0.04). Thus, at the Boston

site, additional lateral force-resisting systems are not required for

the gravity-designed PT flat plate frames, irrespective of the

existence of slab bottom reinforcement.

(a) (b)

Punching PunchingPlastic hinge Plastic hinge

With bottom bar With bottom barWithout bottom bar Without bottom bar

Three-storey Three-storey

Six-storey Six-storey

Nine-storey Nine-storey

13 8 9 14

7 15 1 2 10

11 5 3 6 4 12

19 18

22 14 9 17

21 20 5 23 6 24

15 16 1 12 4 10

11 8 2 13 3 7

30 29

28 24 23 26

27 22 25 9 19 21

20 11 7 8 5 18

14 10 1 3 6 16

15 12 2 4 13 17

5

4 6 1 2 3

13 17 8 18 9 14

7 15 1 16 2 10

11 5 3 6 4 12

15 13 14 16

11 9 1 10 4 12

5 7 2 8 3 6

30 29 32

28 31 19 33 18 25

22 26 14 27 9 17

21 20 5 23 6 24

15 16 1 12 4 10

11 8 2 13 3 7

20 19 21

16 14 5 15 17 18

8 7 1 3 6 9

11 13 2 4 10 12

38 37

36 34 32 33 31 35

30 29 25 24 23 27

28 22 26 9 19 21

20 11 7 8 5 18

14 10 1 3 6 16

15 12 2 4 13 17

Figure 15. (a) Locations of plastic hinges and punching shear

failure at a drift ratio for LS. (b) Locations of plastic hinges and

punching shear failure at a drift ratio for CP

1·2

1·0

0·8

0·6

0·4

0·2

0

Spec

tral

res

pons

e ac

cele

ratio

n: g

0 1 2 3 4Period: s

SDC E, F

SDC D

SDC C

SDC B

SDC A

SDC ESDC D

SDC B

Boston

Seattle

LA

Figure 16. Seismic design category (SDC) at LA, Seattle and

Boston sites

330



Seattle

Figure 18(b) shows vertical distributions of median interstorey

drift ratios through the frames under SAC Seattle 10/50 and 2/50

ground motions. Table 4 summarises Łmax for the frames. For

Seattle 10/50 ground motions, Łmax of all the frames having slab

bottom reinforcement does not exceed the limiting drift ratio for

LS. For the three- and nine-storey frames without slab bottom

reinforcement under 10/50 ground motions, Łmax does not exceed

the limiting drift ratio for LS, whereas Łmax for the six-storey

frame exceeds the limiting drift ratio for LS at the fifth and sixth

storeys.

For Seattle 2/50 ground motions, Łmax for all the frames exceeds

the limiting drifts for LS, irrespective of the existence of slab

bottom reinforcement. For the 2/50 ground motions, Łmax for the

frames having slab bottom reinforcement does not exceed the

limiting drift ratio for CP, whereas Łmax for all frames without

slab bottom reinforcement is greater than the limiting drift for

CP. Thus, to satisfy the basic safety objective (BSO) specified in

FEMA-273 (FEMA, 1997), additional lateral force-resisting

systems are required for the gravity load frames without slab

bottom reinforcement passing through the column.

As summarised in Table 4, for Seattle 10/50 ground motions,

Łmax for the three-, six- and nine-storey frames without slab

bottom reinforcement is 1.21, 1.29 and 1.08 times larger, respec-

tively, than Łmax of the corresponding frames having slab bottom

reinforcement. For 2/50 ground motions, Łmax for the three-storey

frames without slab bottom reinforcement is 1.84 times larger

than that for the corresponding frames having slab bottom

10/50 Hazard level(Scale 0·90)

LA�


LA�


Seattle�

2/50 Hazard level(Scale 0·77)�

Seattle


Boston�

2/50 Hazard level(Scale 0·70)�

Boston

Three-storey Three-storeySix-storey Six-storeyNine-storey Nine-storey4·0

4·0

3·2

3·2

2·4

2·4

1·6

1·6

0·8

0·8

S a: g

1·0

0·8

0·6

0·4

0·2

0

Tn: s0 1 2 3 4 1 2 3 4

Average spectrum Average spectrum

Average spectrum

Average spectrum

Average spectrum

Average spectrum

FEMA 273 spectrum(Site D)






Figure 17. Response spectrum for ground motions at LA, Seattle

and Boston sites

331



3

2

1

06

5

4

3

2

1

09876543210

0 00·01 0·010·02 0·020·03 0·030·04 0·04

10/50 median (with bottom bar) 10/50 median (without bottom bar) 2/50 median (with bottom bar) 2/50 median (without bottom bar)

LS0·

02 r

ad�

LS0·

02 r

ad�

CP

�0·

04 r

ad

CP

�0·

04 r

ad

10/50 Hazard level 2/50 Hazard level3

2

1

06

5

4

3

2

1

09876543210

0 00·01 0·020·02 0·040·03 0·060·04 0·08

LS0·

02 r

ad�

LS0·

02 r

ad�

CP

�0·

04 r

ad

CP

0·04

rad

�

10/50 Hazard level 2/50 Hazard level

Stor

eySt

orey

Stor

ey

Stor

eySt

orey

Stor

ey

Storey drift ratio: rad Storey drift ratio: rad

3

2

1

06

5

4

3

2

1

09876543210

0 00·02 0·020·04 0·040·06 0·060·10 0·10

LS0·

02 r

ad�

LS0·

02 r

ad�

CP

�0·

04 r

ad

CP

0·04

rad

�

10/50 Hazard level 2/50 Hazard level

Stor

eySt

orey

Stor

ey

Storey drift ratio: rad

(a) (b) (c)

0·08 0·080·12 0·12

Figure 18 (a) Median storey drift ratio of three-, six-, and nine-storey PT flat plate frames at Boston site. (b) Median storey drift ratio of three-, six- and nine-storey PT flat plate

frames at Seattle site. (c) Median storey drift ratio of three-, six- and nine-storey PT flat plate frames at LA site

332

Mag

azin

eo

fC

on

crete

Rese

arch

Volu

me

64

Issue

4Effe

cto

fsla

bb

otto

mre

info

rcem

en

to

nse

ismic

perfo

rman

ceo

fp

ost-te

nsio

ned

flat

pla

tefra

mes

Han

,M

oon

and

Park

reinforcement. Unfortunately, for 2/50 ground motions, values of

Łmax of six- and nine-storey frames without slab bottom rein-

forcement become without bound, which indicates that the frames

reach a global dynamic instability state.

LA site

For 10/50 ground motions, median interstorey drift ratios of the

frames without slab bottom reinforcement are much larger than

those of the corresponding frames having slab bottom reinforce-

ment. As shown in Figure 18(c), for 10/50 ground motions, Łmax

for the frames having slab bottom reinforcement slightly exceeds

the limiting drift ratio for LS but does not exceed the limiting

drift for CP. In contrast, for the frames without slab bottom

reinforcement, under 10/50 ground motions Łmax exceeds the

limiting drift ratios for LS as well as CP.

For 2/50 ground motions, Łmax for all the frames exceeds the

limiting drift ratio for CP. Łmax for the three-, six- and nine-storey

frames with slab bottom reinforcement is 0.062, 0.058 and

0.1 rad, respectively, whereas that of the corresponding frames

without slab bottom reinforcement is infinite (see Table 4). If

additional lateral resisting systems are not used with gravity-load-

designed PT flat plate frames located at the LA site, the PT flat

plate frames would experience significant damage and collapse

during earthquakes.

ConclusionsIn this study non-linear static pushover analyses as well as non-

linear response history analyses were conducted to evaluate the

seismic performance of three-, six- and nine-storey gravity-load-

designed PT flat plate frames with or without slab bottom

reinforcement passing through the columns. For this purpose, an

analytical model for PT slab–column connections was developed

which can emulate the actual hysteretic behaviour of PT slab–

column connections and failure mechanisms without the numer-

ical convergence problem caused by interaction among non-linear

springs at the connection. The followings conclusions can be

drawn.

(a) From the non-linear static pushover analyses, strength and

drift capacities of PT flat plate frames are significantly

affected by the slab bottom reinforcement. By placing slab

bottom reinforcement, the lateral strength and drift capacity

of the PT flat plate frames are significantly increased.

(b) At limiting drift ratios of 0.02 and 0.04 corresponding to LS

and CP, respectively, the PT flat plate frames without slab

bottom reinforcement experience more punching shear

failures than the frames with slab bottom reinforcement.

(c) At the Boston site (low seismic site, SDC B), gravity-load-

designed PT flat plate frames satisfy the basic safety

objective (BSO) irrespective of slab bottom reinforcement

passing through the column. Thus, an additional lateral force-

resisting system is not necessary for the gravity-load-

designed PT flat plate frames.

(d ) At the Seattle site (medium seismic site, SDC D), for 10/50Build

ing

Max

imum

store

ydrift

ratio:

rad

10/5

0(m

edia

n)

2/5

0(m

edia

n)

With

bott

om

bar

(1)

Without

bott

om

bar

(2)

(2)

(1)

With

bott

om

bar

(3)

Without

bott

om

bar

(4)

(4)

(3)

Bost

on

Seat

tle

LABost

on

Seat

tle

LABost

on

Seat

tle

LABost

on

Seat

tle

LABost

on

Seat

tle

LABost

on

Seat

tle

LA

Thre

e-st

ore

y0. 0

03

0. 0

14

0. 0

25

0. 0

04

0. 0

17

0. 0

51

1. 3

31. 2

12. 0

40. 0

08

0. 0

32

0. 0

62

0. 0

09

0. 0

59

1. 4

56

1. 1

31. 8

423. 4

8

Six-

store

y0. 0

02

0. 0

17

0. 0

30

0. 0

02

0. 0

22

0. 3

13

1. 0

01. 2

910. 4

30. 0

06

0. 0

27

0. 0

58

0. 0

06

0. 2

87

0. 7

15

1. 0

010. 6

312. 3

2

Nin

e-st

ore

y0. 0

02

0. 0

12

0. 0

27

0. 0

02

0. 0

13

0. 1

30

1. 0

01. 0

84. 8

40. 0

05

0. 0

22

0. 0

99

0. 0

05

0. 2

63

0. 2

06

1. 0

011. 9

52. 0

8

Tab

le4.Su

mm

ary

for

max

imum

store

ydrift

ratio,Ł m

ax,

of

thre

e-,

six-

and

nin

e-st

ore

yfr

ames

atLA

site

333



ground motions, PT flat plate frames having slab bottom

reinforcement satisfy the BSO, whereas PT flat plate frames

without slab bottom reinforcement do not satisfy the BSO.

Thus, additional lateral force-resisting systems would be

required for the gravity-designed PT flat plate frames without

slab bottom reinforcement.

(e) At the LA site (high seismic site, SDC E), the gravity-designed

PT flat plate frames do not satisfy the BSO, irrespective of slab

bottom reinforcement. Thus, an additional lateral force-

resisting system is required to satisfy the BSO irrespective of

slab bottom reinforcement passing through the column.

AcknowledgementsThe authors acknowledge the financial support provided by the

National Research Foundation of Korea (2009-0086384) and

SRC/ERC (R11-2005-0049733). The views expressed are those

of the authors, and do not necessarily represent those of the

sponsors.

REFERENCES

ACI Committee 318 (2005) Building Code Requirements for

Reinforced Concrete. American Concrete Institute,

Farmington Hills, Michigan, ACI 318-05.

ACI–ASCE Joint Committee 352 (1997) Recommendation for

Design of Slab–Column Connections in Monolithic

Reinforced Concrete Structures. American Concrete Institute,

Farmington Hills, Michigan, ACI 352.1R-89.

Banchik CA (1987) Effective Beam Width Coefficients for

Equivalent Frame Analysis of Flat-Plate Structures. ME

thesis, Department of Civil and Environmental Engineering,

University of California at Berkeley, California.

Bracci JM, Reinhorn AR and Mander J (1995) Seismic resistance

of reinforced concrete frame structures designed for gravity

loads: performance of structural system. ACI Structural

Journal 92(5): 597–609.

FEMA (Federal Emergency Management Agency) (1997) NEHRP

Guidelines for the Seismic Rehabilitation of Buildings.

Applied Technology Council, Redwood City, California,

FEMA-273.

Han SW, Kwon OS and Lee LH (2004) Evaluation of the seismic

performance of a three-story ordinary moment-resisting

concrete frame. Earthquake Engineering and Structural

Dynamics 33(6): 669–685.

Han SW, Kee S-H, Kang TH-K et al. (2006a) Cyclic behaviour of

interior post-tensioned flat plate connections. Magazine of

Concrete Research 58(10): 699–711.

Han SW, Kee S-H, Park Y-M, Lee L-H and Kang TH-K (2006b)

Hysteretic behavior of exterior post-tensioned flat plate

connections. Engineering Structures 28(14): 1983–1996.

Han SW, Kee S-H, Ha S-S and Wallace JW (2009) Effects of

bottom reinforcement on hysteretic behavior of post-

tensioned flat plate connections. Journal of Structural

Engineering, ASCE 135(9): 1019–1033.

Hognestad E, Hanson NW and McHenry D (1995) Concrete stress

distribution in ultimate strength design. ACI Structural

Journal 52(4): 455–480.

International Code Council (2006) International Building Code

2006. International Code Council, Whittier, California.

Kang THK and Wallace JW (2005) Dynamic responses of flat

plate systems with shear reinforcement. ACI Structural

Journal 102(5): 763–773.

Martinez-Cruzado JA (1993) Experimental Study of Post-

Tensioned Flat Plate Exterior Slab–Column Connections

Subjected to Gravity and Biaxial Loading. PhD thesis,

Department of Civil Engineering, University of Berkeley,

California.

Midas IT (2004) Midas/GENw User’s Manual (Version 6.3.2).

Midas IT, Seoul, Korea.

Mitchell D and Cook WD (1984) Preventing progressive collapse

of slab structures. Journal of Structural Engineering, ASCE

110(7): 1513–1532.

OpenSees (2007) Open System for Earthquake Engineering

Simulation (OpenSees), Pacific Engineering Research Center,

University of California, Berkeley, California, USA. See

http://opensees.berkeley.edu (accessed 15/09/2011).

Qaisrani AN (1993) Interior Post-Tensioned flat-Plate

Connections Subjected to Vertical and Biaxial Lateral

Loading. PhD thesis, Department of Civil Engineering,

University of California at Berkeley, California.

Spacone E, Filippou FC and Taucer FF (1996) Fiber beam–column

model for nonlinear analysis of R/C frames: Part I.

Formulation. Earthquake Engineering and Structural

Dynamics 25(7): 711–725.

Yun S-Y, Hanmbuger RO, Cornell CA and Foutch DA (2002)

Seismic performance evaluation for steel moment frames.

Journal of Structural Engineering ASCE 128(4): 534–545.

WHAT DO YOU THINK?

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