effect of slab bottom reinforcement on seismic performance...
TRANSCRIPT
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
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Magazine of Concrete ResearchVolume 64 Issue 4
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
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Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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
Ee�
μ /� e emax 1p
(b)
Figure 3. Hysteretic model for PT slab–column connections
(OpenSees, 2007): (a) pinching parameter; (b) unloading
parameter
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Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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
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Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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
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Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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
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Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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)
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Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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
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Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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
Magazine of Concrete ResearchVolume 64 Issue 4
Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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
Magazine of Concrete ResearchVolume 64 Issue 4
Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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
Without bottom bar (4) 908 0.010 0.017
(3)=(4) 1.89 2.50 2.35
Nine-storey With bottom bar (5) 2303 0.020 0.036
Without bottom bar (6) 1086 0.009 0.015
(5)=(6) 2.12 2.22 2.40
Table 3. Summary of maximum base shear and maximum roof
drift ratio
328
Magazine of Concrete ResearchVolume 64 Issue 4
Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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
Magazine of Concrete ResearchVolume 64 Issue 4
Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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
Magazine of Concrete ResearchVolume 64 Issue 4
Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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�
2/50 Hazard level(Scale 0·83)
LA�
10/50 Hazard level(Scale 0·73)
Seattle�
2/50 Hazard level(Scale 0·77)�
Seattle
10/50 Hazard level(Scale 0·63)
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)
FEMA 273 spectrum(Site D)
FEMA 273 spectrum(Site D)
FEMA 273 spectrum(Site D)
FEMA 273 spectrum(Site D)
FEMA 273 spectrum(Site D)
Figure 17. Response spectrum for ground motions at LA, Seattle
and Boston sites
331
Magazine of Concrete ResearchVolume 64 Issue 4
Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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
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crete
Rese
arch
Volu
me
64
Issue
4Effe
cto
fsla
bb
otto
mre
info
rcem
en
to
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ismic
perfo
rman
ceo
fp
ost-te
nsio
ned
flat
pla
tefra
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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
Magazine of Concrete ResearchVolume 64 Issue 4
Effect of slab bottom reinforcement onseismic performance of post-tensionedflat plate framesHan, Moon and Park
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.
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