Earthquake Response Analysis of Multiple Towers On
Earthquake Response Analysis of Multiple Towers On
Earthquake Response Analysis of Multiple Towers On
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Experimental Evaluation of the Lateral Load Behavior of Squat Structural Walls View project
Seismic Retrofit of Reinforced Concrete Shear Walls Designed and Constructed Prior to Enforcement of the Recent Seismic Design Codes View project
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Cem TURA1
Kutay ORAKÇAL2
ABSTRACT
In this study, a hypothetical tall building structure consisting of two towers and a common
podium is analyzed, considering effects of interaction between the towers due to the
connected podium floors. Interaction effects on the podium level are included in the analyses
using either an upper bound or lower bound approach, by assigning fixed or free end restraints
at the continuous boundaries of single tower models, respectively. Responses of the single
tower models are then compared with the response obtained for the combined model, which
includes both of the towers as well as the common podium and basements. Results obtained
using different linear and nonlinear analysis methods indicate that single tower models with
fixed end restraints overestimate the internal forces at the podium floors, although to a
reasonable extent.
Keywords: Tall building, multiple towers, common podium, nonlinear analysis,
performance based design.
1. INTRODUCTION
In projects with multiple towers on a common podium, even when the structural properties
of the towers are similar to each other, dynamic response of the towers during real seismic
events are likely to vary due to several reasons, including the service loads on the towers,
local soil conditions, and spatial variability of the ground motion. Possible out of phase
response of individual towers due to this variance in dynamic response may create excessive
in-plane stresses at the connected podium level diaphragms. To prevent possible diaphragm
failure, design engineers must consider these effects as critical (i.e., non-ductile) design
quantities. Furthermore, such interaction between the towers may also influence the response
Note:
- This paper has been received on October 04, 2018 and accepted for publication by the Editorial Board
on January 28, 2019.
- Discussions on this paper will be accepted by January 31, 2020.
https://dx.doi.org/10.18400/tekderg.467371
parameters (e.g., interstory drifts, story shear forces, story overturning moments, etc.)
associated with the design of each individual tower.
Contrary to linear elastic analysis methods and strength based design used for regular
structures, nonlinear response history analysis is required for tall buildings within the
performance based design framework. Nevertheless, in real-life practice, due to time
restrictions on project schedules, analysis of multiple towers using a combined structural
model is typically not employed, and interaction effects between multiple towers are simply
neglected, since computational demand for nonlinear response history analyses of multiple-
tower models is significantly high. However, diaphragm forces developing in the podium
floors of such structures is not an issue that can be disregarded in design.
Multiple towers on common podium buildings were not prevalent up to recent years since
such structures were designed using seismic joints, forcing the response of the towers to be
independent of each other. Therefore, although there is a scarce amount of relevant research
in the literature, analysis of multiple towers on a common base is a subject that is not yet
adequately investigated. One of the few studies that focus on multiple towers on common
base is conducted by Qi and Chen [1]. Dynamic behavior of two towers connected with a
typical podium structure is investigated through a parametric study. Towers and podium
levels are modeled linear elastically using a lumped mass and equivalent stiffness approach.
A simplified three degree of freedom (DOF) system is analyzed under a “constant”
acceleration response spectrum, considering variation in mass and stiffness characteristics.
This early study identifies the interaction effects between the towers to the seismic response
of each, considering linear elastic forces only. However, this study does not include
investigation of diaphragm effects at the connecting podium levels. In a more recent study
by Behnamfar et al. [2], the importance of interaction effects between the two structures with
various dynamic properties are addressed. A formulation is developed for simple multi
degree of freedom systems to examine the severity of interaction forces on each structure.
Linear springs are suggested for analysis using separate models for each tower. However,
these linear springs are intended to represent only the kinematic interaction between the
towers and neglect kinetic interaction caused by inertial effects. In addition, defining stiffness
values for these springs may be tedious in real projects with complex structural
configurations, especially projects that contain more than two towers on a single podium.
Furthermore, in high rise structures, kinetic (i.e., inertial) interaction caused by out-of-phase
vibration of the towers can be more predominant compared to kinematic interaction.
Although this study provides notable amount of information for structures where inertial
interaction is not expected to significantly affect the structural response, it is not adequate for
reliable seismic response analysis of structures with high rise towers and connecting
podiums.
The dynamic interaction between the towers and its effect on the dynamic response of the
overall structure and the diaphragm effects on the podium levels is an issue that is explicitly
warned against in design guidelines [3, 4, 5]; however, extremely limited literature is
available on the subject. There is a clear need for additional analytical studies on this topic,
which focus on the nonlinear response of real-life structural systems. As well, considering
computational limitations, there is also a need for development of more practical approaches
to be used for analysis and design of such structures.
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Therefore, in this study, a practical approach is sought to evaluate the seismic response of
structures incorporating multiple towers on a single podium, with emphasis on assessment of
diaphragm effects at the podium levels. The main motivation is to assess whether critical
response quantities for the towers and the podium floors can be approximately obtained
within less computation time, by analyzing the towers separately with different assumptions
for the end restraints, instead of using a combined model for analysis of the multiple tower
structure.
Within the scope of this work, linear and nonlinear models of a hypothetical structure with
two towers on a common podium are generated. The structure is analyzed using single-tower
and combined double-tower models. The combined double model incorporates the entire
structure including the towers, as well as the connecting podium and basement levels. The
single models consist of a single individual tower, together with the corresponding half of
the podium and basement floors, with different kinematic boundary conditions defined at the
interface.
2. METHODOLOGY
2.1. Building Properties
The hypothetical reinforced concrete structure investigated in this study consists of two
towers with 44 stories above and five stories below ground level. With 3.2 m typical height
for each story, total height of the structure is 156.8 m and clear heights of the towers are
140.8 m. An identical structural system for both towers consist of a core wall system
connected with coupling beams, T-shaped walls, outrigger beams (connecting core walls to
perimeter walls), and perimeter frames along all four edges. The two towers are connected to
each other by four podium floors above ground level, as well as five basement floors
surrounded by perimeter walls (Figure 1). Architectural properties of the building are
assumed such that the basement floors are utilized as parking garages, podium floors are
utilized as commercial zones, and towers are utilized as residential buildings. Based on
strength based design of the structure compliant with Turkish Building Earthquake Code
2018 (TBEC2018) [6], cross-sectional dimensions of the structural members are obtained as
presented in Table 1.
Figure 1 - 3D view of the structure and typical plan view for connected podium floors.
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Earthquake Response Analysis of Multiple Towers on a Common Podium
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Earthquake Response Analysis of Multiple Towers on a Common Podium
Figure 3 - Mode shapes for fundamental periods (a) TBY, (b), TAX, (c) TAY, (d) TBX.
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Earthquake Response Analysis of Multiple Towers on a Common Podium
Figure 5 - Modeling of frame elements with lumped plasticity approach in Perform 3D; (a)
coupling beams (b) outrigger and perimeter beams (c) columns.
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3.0
DD1 Level
DD2 Level
2.5
1.5
1.0
0.5
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Period (s)
10 10
SRSS Spectra SRSS Spectra
1.3 DD1 Level Spectrum 1.3 DD2 Level Spectrum
Average Spectrum Average Spectrum
Spectral Acceleration (g)
1 1
0.1 0.1
0.01 0.01
0.75 2.25 3.75 5.25 6.75 8.25 0 1 2 3 4 5 6
Period (s) Period (s)
(a) (b)
Figure 8 - SRSS acceleration spectra of selected ground motions for earthquake levels;
(a) DD1 (b) DD2.
3. ANALYSIS RESULTS
In linear analyses, design spectrum of DD2 level (design level, i.e., 10% probability of
exceedance in 50 years) seismicity (Figure 7) is used for RSA and 11 pairs of ground motion
records selected for the target spectrum of DD2 level seismicity (Figure 8) are used for
LMTHA. Furthermore, seismic demand is defined for linear analyses using a structural
system behavior factor R 6 for structural systems where all seismic demand is resisted by
uncoupled structural walls according to TBEC2018. On the other hand, 11 pairs of ground
motion records selected for the design spectrum corresponding to DD1 level (maximum
credible, i.e., 2% probability of exceedance in 50 years) seismicity are used for NLRHA. In
all response history analyses, total of 22 analyses are conducted, first by applying 11 ground
motion record pairs, and secondly by reapplying the records at 90 degree rotated state. In all
analyses, in-plane (diaphragm) tensile forces in X direction and in-plane (diaphragm) shear
forces are obtained at section cuts defined adjacent to the tower edges (Figure 2). During
evaluation of analysis results of linear analysis, earthquake effects are magnified using the
over-strength factor D 2.5, as is specified in TBEC2018.
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Earthquake Response Analysis of Multiple Towers on a Common Podium
Figure 9 - Diaphragm tensile force distribution (in X direction) per unit length of slab at
+12.80 m elevation of double model obtained from RSA.
Resultant tensile forces obtained from RSA of the double and single models, developing in
the connected podium diaphragms are presented in Figure 10. As it is expected, interaction
effects are much more critical at the uppermost diaphragm level compared to those at lower
stories. When the resultant tensile forces at the section cuts are evaluated, it is clear that single
models with fixed end restraints overestimate the in-plane axial forces at the connected
podium slabs. According to section cut results obtained from RSA of the double model,
average tensile stresses (averaged over the entire slab cross section) are calculated as 2.68
MPa for Section A and 2.75 MPa for Section B of the diaphragm, at +12.80 m elevation. In
addition, axial loads on beams range between 800 kN and 1000 kN, with a total of 4420 kN
at Section A and 3540 kN for Section B. Considering this distribution, additional diaphragm
reinforcement can be designed as 16/300 mm (1340 mm2/m) additional top and bottom
reinforcement in slabs and 8 (2512 mm2) skin (i.e., longitudinal web) reinforcement in
beams at Section A, and 16/300 mm (1340 mm2/m) additional top and bottom reinforcement
in slabs and 8 (2512 mm2) skin reinforcement in beams at Section B. Furthermore, when
average stresses at lower podium levels are evaluated (1.36 MPa for Section A, 1.41 MPa for
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Section B), it is noted that concrete at these levels does not crack under axial tension;
therefore, no additional reinforcement in the slab or the beams is required.
14.4 14.4
12.8 12.8
11.2 11.2
Elevation (m)
Elevation (m)
9.6 9.6
8.0 8.0
6.4 6.4
4.8 4.8
3.2 Double Model
3.2
Single Model
1.6 1.6
0 10000 20000 30000 40000 0 5000 10000 15000 20000 25000 30000
Tensile Force (kN) Tensile Force (kN)
(a) (b)
Figure 10 - Resultant tensile forces (in X direction) obtained from RSA at (a) Section A,
(b) Section B.
On the other hand, if the single-fixed model analysis results are used in design, additional
slab and beam reinforcement amounts calculated using the double model increase up to
16/200 mm (2010 mm2/m) and 8 (4248 mm2) at Section A, and 16/175 mm (2297
mm2/m) and 8 (4248 mm2) at Section B. Furthermore, according to the single-fixed model
results, the diaphragm at +09.60 m elevation also cracks and it can be designed for 12/200
mm (1130 mm2/m) additional reinforcement in the slabs and 6 (2280 mm2) skin
(longitudinal web) reinforcement in beams at Section A, and 12/175 mm (1291 mm2/m)
additional reinforcement in the slabs and 8 (2512 mm2) skin reinforcement in the beams
at Section B.
In-plane distribution of the resultant shear forces (per unit slab length) developing in the
podium floor diaphragm at +12.80 m elevation is presented in Figure 11, based on RSA
results using the double model. Resultant shear forces in the slab elements vary between 200
kN/m and 250 kN/m in the regions between the two towers. Comparing these magnitudes
with a resultant shear force capacity 214 kN/m (corresponding to the concrete shear strength
of 1.07 MPa), it is observed that diaphragm shear forces are as not critical as tensile forces
for this structure. Furthermore, resultant shear forces in the connected diaphragms are
presented in Figure 12 for RSA using double and single-fixed models. Similar to the tensile
force distributions, total shear forces are highest at the uppermost connecting diaphragm,
with magnitudes of 7046 kN at Section A and 5542 kN at Section B, and reduce throughout
the lower stories down to 3785 kN for Section A and 2778 kN for Section B. Besides,
resultant shear forces obtained using double and single-fixed models differ significantly,
similarly to tensile force resultants. Therefore, if analysis results using the single-fixed model
are used in design, average shear stress values increase up to 2.27 MPa for Section A and
2.41 MPa for Section B. Accordingly, the required amount of additional slab reinforcement
for design increases up to 658 mm2/m for Section A and 734 mm2/m for Section B. Based on
these demands, additional slab reinforcement can be designed as 12/350 mm (646 mm2/m)
top and bottom bars at Section A, and 12/300 mm (753 mm2/m) top and bottom bars at
Section B. Furthermore, according to section cut results of the fixed model under RSA,
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Earthquake Response Analysis of Multiple Towers on a Common Podium
average shear stresses in the diaphragm at +09.60 m elevation are calculated as 1.29 MPa at
Section A and 1.35 MPa at Section B. Since these average stress levels are very close to the
concrete design shear strength of 1.07 MPa, the required amount of additional reinforcement
can also considered to be negligible for design purposes.
Figure 11 - Diaphragm shear force distribution per unit length of slab at +12.80 m
elevation of double model obtained from RSA.
14.4 14.4
12.8 12.8
11.2 11.2
Elevation (m)
Elevation (m)
9.6 9.6
8.0 8.0
6.4 6.4
4.8 4.8
3.2 3.2 Double Model
Single Model
1.6 1.6
0 5000 10000 15000 20000 0 2000 4000 6000 8000 10000 12000 14000
Shear Force (kN) Shear Force (kN)
(a) (b)
Figure 12 - Resultant diaphragm shear forces obtained from RSA at (a) Section A,
(b) Section B.
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similar resultant tensile force magnitudes and distributions compared to RSA results. The
only difference noted is marginally smaller magnitudes at critical regions of the diaphragm.
Resultant tensile forces in the connected podium floors, obtained from LMTHA of the
structure using the double and single-fixed models, is presented in Figure 14 For consistency
with RSA results, mean of 22 analysis results are presented in the table and figure. Similar
to RSA, analysis results of the single-fixed model overestimate the double model results.
Based on the average of 22 analyses, maximum resultant tensile force values obtained in the
first connecting diaphragm are 18024 kN at Section A and 14322 kN at Section B. These
resultants corresponds to average tensile stresses of 2.50 MPa at Section A and 2.63 MPa at
Section B. Since these stress levels are close to results of RSA, an identical reinforcement
design is applicable to resist the diaphragm forces.
Figure 13 - Diaphragm tensile force distribution (in X direction) per unit length of slab at
+12.80 m elevation of double model obtained from LMTHA.
In-plane distribution of resultant shear forces in the floor diaphragm at +12.80 m elevation is
presented in Figure 15 for LMTHA using the double model. Similar to tensile forces,
distribution of resultant shear forces are also similar to RSA results. As was the case in RSA,
resultant shear forces (per unit length) in the slabs vary between 200 kN/m and 250 kN/m in
the region between the two towers.
Resultant shear forces along the connected podium floors are presented in Figure 16 for
LMTHA using single-fixed and double models. Similar to RSA, analysis results obtained
using the single- fixed model overestimate the double model results. According to average
of 22 analyses, maximum resultant shear forces (per unit slab length) developing in the
topmost connecting diaphragm are 5359 kN at Section A and 4324 kN at Section B. These
resultants correspond to average shear stresses of 0.84 MPa at Section A and 0.90 MPa at
Section B. Since these shear stress levels do not exceed the concrete design shear strength of
1.07 MPa, no additional diaphragm shear reinforcement is required.
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Earthquake Response Analysis of Multiple Towers on a Common Podium
14.4 14.4
12.8 12.8
11.2 11.2
Elevation (m)
Elevation(m)
9.6 9.6
8.0 8.0
6.4 6.4
Double Model
4.8 4.8 Single Model
3.2 3.2 Double Model Average
Single Model Average
1.6 1.6
0 10000 20000 30000 40000 50000 60000 0 10000 20000 30000 40000
Tensile Force (kN) Tensile Force (kN)
(a) (b)
Figure 14 - Resultant tensile forces (in X direction) obtained from LMTHA at (a) Section A,
(b) Section B.
Figure 15 - Diaphragm shear force distribution per unit length of slab at +12.80 m
elevation of double model obtained from LMTHA.
On the other hand, if single-fixed model analysis results are used in design, average shear
stresses increase up to 2.27 MPa at Section A and 2.18 MPa at Section B. Since these stresses
exceed the design shear strength of concrete 1.07 MPa, the required amount of additional
slab reinforcement increases up to 658 mm2/m at Section A and 608 mm2/m at Section B at
this level. According to these amounts, additional slab reinforcement can be designed as
12/350 mm (646 mm2/m) top and bottom bars at Section A, and 12/350 mm (646 mm2/m)
top and bottom bars at Section B. In addition, according to section cut results of the fixed
model under LMTHA, average shear stresses in the diaphragm of +09.60 m elevation are
calculated as 1.34 MPa at Section A and 1.29 MPa at Section B. Since these stress levels are
very close to the concrete design shear strength of 1.07 MPa, the required amount of
additional slab reinforcement is negligible in design.
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14.4 14.4
12.8 12.8
11.2 11.2
Elevation (m)
Elevation (m)
9.6 9.6
8.0 8.0
6.4 6.4
Double Model
4.8 4.8 Single Model
3.2 3.2 Double Model Average
Single Model Average
1.6 1.6
0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000
Shear Force (kN) Shear Force (kN)
(a) (b)
Figure 16 - Resultant diaphragm shear forces obtained from LMTHA at (a) Section A,
(b) Section B.
In the following figures, comparison of resultant tensile and shear forces obtained from RSA
and LMTHA of the single-fixed and double models, throughout the connected podium floors
are presented. According to Figure 17, LMTHA consistently gives smaller tensile force
values with respect to RSA. Resultant tensile forces at critical sections are very close to each
other at the topmost connecting diaphragm, except for the single model results obtained for
Tower B. Furthermore, differences in results of the two analysis methods are more significant
at lower diaphragm levels. Similarly, according to Figure 18, diaphragm shear force
resultants obtained by LMTHA are generally smaller than RSA results. The only exception
is the fixed model results for the section cut at +09.60 m elevation of Tower A. At this section,
the result of LMTHA is barely higher than the RSA result.
14.4 14.4
12.8 12.8
11.2 11.2
Elevation (m)
Elevation (m)
9.6 9.6
8.0 8.0
6.4 6.4
Double Model RSA
4.8 4.8 Single Model RSA
3.2 3.2 Double Model LMTHA
Single Model LMTHA
1.6 1.6
0 5000 10000 15000 20000 25000 30000 35000 0 5000 10000 15000 20000 25000 30000
Tensile Force (kN) Tensile Force (kN)
(a) (b)
Figure 17 - Comparison of resultant diaphragm tensile forces (in X direction) obtained
from LMTHA and RSA at (a) Section A, (b) Section B.
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Earthquake Response Analysis of Multiple Towers on a Common Podium
14.4 14.4
12.8 12.8
11.2 11.2
Elevation (m)
Elevation (m)
9.6 9.6
8.0 8.0
6.4 6.4
Double Model RSA
4.8 4.8 Single Model RSA
3.2 3.2 Double Model LMTHA
Single Model LMTHA
1.6 1.6
0 5000 10000 15000 20000 0 2000 4000 6000 8000 10000 12000 14000
Shear Force (kN) Shear Force (kN)
(a) (b)
Figure 18 - Comparison of resultant diaphragm shear forces obtained from LMTHA and
RSA at (a) Section A, (b) Section B.
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Diaphragm tensile force resultants developing in the connected podium floors are presented
in Figure 20, for the average of 22 NLRHA cases (corresponding to 22 ground motions).
Compared to results of RSA and NLRHA, force resultant distribution along the podium floor
elevations follows a similar pattern. However, there is 18% increase at Section A and 31%
increase at Section B, in the total tensile force resultants at the floor with +12.80 m elevation,
compared to RSA. This percent difference reaches upto 48% increase at Section A and 35%
increase at Section B, when analysis results using the single-fixed models are considered.
Figure 19 - Diaphragm tensile force distribution (in X direction) per unit length of slab at
+12.80 m elevation of double model obtained from NLRHA under record RSN1762 [14].
Considering mean analysis results obtained using the double model, average tensile stresses
developing at the topmost podium diaphragm are calculated as 3.17 MPa at Section A and
3.60 MPa at Section B. Comparing these stress levels with the expected tensile strength of
concrete 2.82 MPa, it is deduced that concrete cracks in tension and additional reinforcement
is required in these regions, similar to linear analysis results. Using the expected yield
strength of reinforcement, total amounts of required slab reinforcement are calculated as
45339 mm2 at Section A and 38813 mm2 at Section B.
Furthermore, contribution of beams to the total tensile force resultant is obtained as 6762 kN
at Section A and 5243 kN at Section B. Note that, percentages corresponding to contribution
of beams to the total diaphragm tension force are 30% for Section A and 27% for Section B,
whereas they were obtained as 23% for Section A and 24% for Section B from RSA. Taking
into account all the information mentioned above, the required additional reinforcement can
be designed as 14/300 mm (1027 mm2/m) top and bottom bars in the slabs and 920 (2826
mm2) skin (i.e., longitudinal web) reinforcement in the beams at Section A, and14/250 mm
(1232 mm2/m) top and bottom bars in slabs and 722 (2660 mm2) skin reinforcement in the
beams at Section B. Average tensile stress levels at lower podium floors, which are 1.79 MPa
at Section A and 1.88 MPa at Section B, are lower than the expected tensile strength of
concrete. Therefore, no additional reinforcement is required to resist diaphragm tension at
lower podium floors.
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Earthquake Response Analysis of Multiple Towers on a Common Podium
14.4 14.4
12.8 12.8
11.2 11.2
Elevation (m)
Elevation (m)
9.6 9.6
8.0 8.0
6.4 6.4
Double Model
4.8 4.8 Single Model
3.2 3.2 Double Model Average
Single Model Average
1.6 1.6
0 20000 40000 60000 80000 0 10000 20000 30000 40000 50000 60000 70000
Tensile Force (kN) Tensile Force (kN)
(a) (b)
Figure 20 - Resultant tensile forces (in X direction) obtained from NLRHA at (a) Section A,
(b) Section B.
On the other hand, when fixed model results are taken into account, average tensile stresses
developing in the slabs at the topmost podium floor increase to 6.70 MPa at Section A and
6.82 MPa at Section B. According to these stresses, the required additional reinforcement
can be designed as 18/250 mm (2032 mm2/m) top and bottom bars in slabs, and 830 (5656
mm2) skin reinforcement in beams at Section A, and 20/300 mm (2093 mm2/m) top and
bottom bars in slabs and 828 (4928 mm2) skin reinforcement in beams at Section B.
Additionally, average tensile stress levels at the podium floor of +09.60 m elevation increases
to 3.41 MPa at Section A and 3.29 MPa at Section B. Therefore, concrete at this floor also
cracks and additional tension reinforcement is required. Required additional reinforcement
at this elevation can be designed as 14/300 mm (1027 mm2/m) top and bottom bars in slabs
and 822 (3040 mm2) skin (i.e., longitudinal web) reinforcement in beams at Section A, and
14/300 mm (1027 mm2/m) top and bottom bars in slabs and 820 (2512 mm2) skin
reinforcement in beams at Section B.
In plane distribution of diaphragm shear forces (per unit length) in the podium floor slab at
+12.80 m elevation is presented in Figure 21 for NLRHA of the double model under the
RSN1762_0 [14] ground motion record (scaled with a factor of 3.326). Similarly to
diaphragm tensile forces, despite crude meshing, diaphragm shear effects are clearly reflected
in the analysis results. The in-plane shear force distribution is similar to linear analysis
results.
Resultant diaphragm shear forces along the connected podium floors are presented in Figure
22, as average of the 22 analysis cases used in NLRHA. Comparing results of RSA (Figure
12, under DD2 level earthquake) and NLRHA (Figure 22, under DD1 level earthquake), the
distribution along the floors follows similar patterns in the single-fixed models, and slightly
different patterns in the double models. In terms of magnitudes, there is a 91% increase at
Section A and an 87% increase at Section B in the total diaphragm shear force acting on the
podium floor with +12.80 m elevation in the double model, compared to RSA. This percent
increase changes to 109% increase at Section A and 81% increase at Section B, when RSA
and NLRHA results using fixed models are considered.
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Figure 21 - Diaphragm shear force distribution per unit length of slab at +12.80 m
elevation of double model obtained from NLRHA under RSN1762 [14] record.
Considering mean response quantities of double model, average diaphragm shear stress
values at the topmost podium floor are calculated as 2.10 MPa at Section A and 2.16 MPa at
Section B. Since these stress levels are very close to the concrete design shear strength 1.83
MPa, the required amounts of additional diaphragm shear reinforcement are negligible
amounts for design purposes. Furthermore, analysis results show that the contribution of
beams to diaphragm shear forces is negligible, as is typical.
14.4 14.4
12.8 12.8
11.2 11.2
Elevation (m)
Elevation (m)
9.6 9.6
8.0 8.0
6.4 6.4
Double Model
4.8 4.8 Single Model
3.2 3.2 Double Model Average
Single Model Average
1.6 1.6
0 10000 20000 30000 40000 50000 0 5000 10000 15000 20000 25000 30000 35000
Shear Force (kN) Shear Force (kN)
(a) (b)
Figure 22 - Resultant shear forces obtained from NLRHA (a) Section A, (b) Section B.
On the other hand, if single-fixed model analysis results are considered in design, average
diaphragm shear stress values at the topmost podium floor increase to 4.76 MPa at Section
A and 4.37 MPa at Section B. Therefore, required amount of additional slab reinforcement
increases up to 1163 mm2/m at Section A and 1008 mm2/m at Section B, at this elevation.
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Earthquake Response Analysis of Multiple Towers on a Common Podium
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Figure 23 - NLRHA interstory drift ratio, structural wall shear force and strain control
nodes for Tower A.
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Earthquake Response Analysis of Multiple Towers on a Common Podium
140 140
120 120
100 100
80 80
Elevation (m)
Elevation (m)
60 60
40 40
20 20
0 0
-20 -20
-0.04 -0.02 0 0.02 0.04 -0.04 -0.02 0 0.02 0.04
Inter-story Drift Ratio Inter-story Drift Ratio
(a) (b)
Figure 24 - Comparison of NLRHA interstory drift ratio distributions for double, single-
fixed and single-free models of Tower A at (a) DA-X, (b) DA-Y.
80 80 80
Elevation (m)
Elevation (m)
Elevation (m)
60 60 60
40 40 40
20 20 20
0 0 0
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80 80 80
Elevation (m)
Elevation (m)
Elevation (m)
60 60 60
40 40 40
20 20 20
0 0 0
80 80 80
Elevation (m)
Elevation (m)
Elevation (m)
60 60 60
40 40 40
20 20 20
0 0 0
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Earthquake Response Analysis of Multiple Towers on a Common Podium
80 80 80
Elevation (m)
Elevation (m)
Elevation (m)
60 60 60
40 40 40
20 20 20
0 0 0
80 80 80
Elevation (m)
Elevation (m)
Elevation (m)
60 60 60
40 40 40
20 20 20
0 0 0
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80 80 80
Elevation (m)
Elevation (m)
Elevation (m)
60 60 60
40 40 40
20 20 20
0 0 0
4. CONCLUSIONS
Under the light of the analysis results obtained using different modeling approaches and
analysis methods used in this study, the following conclusions can be drawn:
When results of linear elastic analysis methods (RSA and LMTHA) conducted under the
design level (DD2) earthquake effects using double (combined double tower) and single-
fixed (single tower with fixed end restraints) model results are compared, it is observed
that RSA provides diaphragm tensile force resultants that are only approximately 10%
larger than LMTHA at the connected podium levels. The difference between results of the
two analysis methods increase to 25% only for diaphragm shear forces obtained using the
double model, yet these diaphragm shear forces are relatively small in magnitude.
When linear elastic analysis results using the double and single-fixed models are
compared, it is observed that the single-fixed model can provide diaphragm tension forces
that are up to 75% higher that the double model results, independently from the analysis
method used (RSA or LMTHA). However, this percent difference increases to more than
100%, when diaphragm shear forces are considered, also because the diaphragm shear
force magnitudes are small.
When results of NLRHA obtained using double and single-fixed models are compared, it
is observed that the single-fixed model can provide diaphragm force resultants that are
almost twice those calculated using the double model.
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NLRHA of the double model of the structure under the DD1 level (maximum credible)
earthquake gives approximately 20-30% higher diaphragm tensile forces compared to
RSA of the double model under the DD2 level (design level) earthquake. On the other
hand, in case of diaphragm shear forces, NLRHA results obtained using the double model
are almost twice those obtained from RSA at critical sections. Although NLRHA gives
higher diaphragm forces, the total amount of required diaphragm reinforcement for tension
is 5% to 15% less than RSA, since expected material strengths are used for design based
on NLRHA results. Additionally, diaphragm shear forces obtained in both analyses do not
exceed concrete shear strength limits, not necessitating any diaphragm shear
reinforcement.
Differently from the double model, when the single-fixed model is used in the analysis,
NLRHA can produce 50% higher diaphragm tensile forces as compared to RSA. On the
other hand, diaphragm shear forces from NLRHA are approximately twice those obtained
by RSA, as observed in the double model. When required reinforcement amounts against
diaphragm tension are compared, NLRHA results using the single-fixed model require
approximately 10% larger amount of reinforcement, as compared to RHA of the single-
fixed model.
Interestingly, the amount of diaphragm reinforcement obtained from NLRHA of the
double model under the DD1 level (maximum credible) earthquake, which is the most
robust and reliable analysis approach, is less than the reinforcement amount obtained from
RSA of the single-fixed model under the DD2 level (design level), which is the simplest
analysis approach that can be used for diaphragm design. This happens mostly because
the fixed model overestimates the diaphragm effects in the connecting podium floors, and
also because the performance-based design based on NLRHA uses expected material
strengths, rather than reduced design strength values for the materials.
NLRHA results show that the single-free (single tower with free end restraints) model
provides results for critical response quantities associated with the seismic performance
of the individual tower structures (interstory drifts, wall strains, wall shear forces, beam
plastic rotations, etc.) that are reasonably close to analysis results obtained using the
double model.
Generally, taking into consideration of analysis duration and modeling complexity, it is
recommended to use single-tower models with free end restraints for design of the
individual towers, and single-tower models with fixed end restraints for design of the
podium slabs for in-plane axial load and shear forces, whenever comprehensive analyses
using a combined multiple-tower model is not possible.
Notations
LMTHA Linear modal time history analysis
NLRHA Nonlinear response history analysis
RSA Response spectrum analysis
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