The document discusses torsion responses in structures due to eccentricity in mass and stiffness distributions and accidental causes such as uncertainties in masses, stiffnesses, and ground motions. Eccentricity is measured as the distance between the center of mass and center of resistance, which causes a torsion moment that must be resisted. Old seismic codes accounted for increased shear from torsion by using a design eccentricity of 1.5 times the actual eccentricity and distributing increased shear but not decreased shear. The literature review discusses reports of damage to asymmetric buildings from earthquakes and how asymmetry causes torsion since the center of mass and center of rigidity do not coincide.
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Torsional response of assymetric multy story building thesis
1. CHAPTER 1
INTRODUCTION
1.1 GENERAL
Torsion responses in structures arise from two sources: Eccentricity in the mass
and stiffness distributions, causing a torsion response coupled with translation response;
and torsion arising from accidental causes, including uncertainties in the masses and
stiffness, the differences in coupling of the structural foundation with the supporting earth
or rock beneath and wave propagation effects in the earthquake motions that give a
torsion input to the ground, as well as torsion motions in the earth itself during the
earthquake. (1)
Horizontal twisting occurs in buildings when the center of mass (CM) does not
coincide with the centre of resistance (CR). The distance between them is called the
eccentricity (e). Lateral force multiplied by this (e) causes a torsion moment (T) that must
be resisted by the structure in addition to the normal seismic force.(2) The centre of
rigidity is the point through which the resultant of the restoring forces of a system acts.
The centre of mass corresponding to centre of gravity (c.g.) of the systems it is the point
through which the resultant of the masses of a system acts. (3)
1.2 THEORY
In general, the torsion arising from eccentric distributions of mass and stiffness
can be taken into account by ascribing an incremental torsion moment (T) in each storey =
the shear (V) in that storey multiplied by the eccentricity (e), measured perpendicular to
the direction of applied ground motion. A precise evaluation of the torsion response is
quite complicated because the coupled lateral- torsion vibration modes of the entire
structure are to be considered by performing a two – or three dimensional response
2. (1)
calculations. As an approximation, the torsion moment in each storey can be obtained
by summing from the top storey the incremental torsion moments.
The “static” torsion responses in each storey are then determined by computing
the twist in each storey obtained by dividing the total torsion storey moment by the storey
“rotational stiffness”. These twists are then added from the base upward to obtain the total
twisting or torsion response at each floor-level.(1) Since these are “static “responses, they
should be “amplified” for “dynamic” response using the response- spectrum amplification
factor for the fundamental torsion frequency of the structure. However, in many design
(1)
codes no amplification whatsoever is used. “Accidental” torsion may arise in many
ways. Most current codes (4) use accidental eccentricity value of 5% of the plan dimension
of the storey perpendicular to the direction of applied ground motion. The accidental
torsion may be considered as an increase and also as a decrease in the eccentricity.
Corresponding to the distance between the centre of mass and resistance in various
storeys; with consideration of increases in all levels or decreases in all levels to get two
bounding values. The accidental torsion (or the total torsion) is computed in the same way
as the “real” torsion described above. (1)
1.3 DISTRIBUTION OF SHEAR AND MOMENTS
The storey shears arising from translation and from torsion response are
distributed over the height of the building in proportion to the stiffness of various
elements in the building the translational shears being affected by the translational
stiffness and the torsion shears being affected by the rotational stiffness of the building. (1)
The computed stiffness of the structure should take into account the stiffness of the floors
of floor structure acting as diaphragm or distributing element. If the floor diaphragm is
(1)
considered as infinitely rigid, and the storey stiffness are of importance. However, if
-2-
3. the floor diaphragm is flexible and deforms greatly, the distribution of forces becomes
more nearly uniform than determined by the method discussed above. A simplified
approach is possible by considering the relative displacements of the building due to
translation and that due to rotation of each storey separately, as affected by the diaphragm
or floor stiffness. The stiff nesses are determined by the forces corresponding to a unit
displacement in either translation or torsion. Respectively, then the shears due to
translation or rotation can be distributed in proportion to these stiff nesses.
The storey moments are distributed to the various frames and walls that make up
the lateral force system in a manner consistent with the distribution of storey shears. In
particular, the shears and moments in any frame or wall should be statically consistent.
Base or “overturning” moments: The flexural base moment is of importance in
connection with the foundation design. The corresponding flexural moments at each floor
level are important in connection with the calculation of vertical stresses in the columns
and walls of the structure. These moments can be computed from modal analysis or
equivalent lateral force analysis.
1.4 OLD CODE PROVISIONS
In 1984 version of Indian Seismic Code makes provision for the increase in shear
resulting from horizontal torsion due to the eccentricity (e) between the centre of mass
and the centre of rigidity. The torsion moment (T) at each storey = the shear (V) in that
storey multiplied by eccentricity (e). Since there could be quite a bit of variation in the
computed value of e, the code recommends that the design eccentricity (ed) be taken as
1.5e. Negative torsion shears shall be neglected. (3)
The net effect of this torsion is to increase the shear in certain structural elements
and reduction in certain others. The code recommends that reduction in shear on account
-3-
4. of torsion should not be applied and only increased shear in the elements be considered.
(2)
The torsion forces shall be distributed to the various vertical components of the seismic
resisting system with due consideration given to the relative stiff nesses of the vertical
components and the diaphragm. It is then corrected for torsion taking into account the
(2) (3)
increases produced, but not the decreases as specified in the code. The following
steps are involved to determine the additional shears due to torsion in a building. Fig. 1.1
Let OX and OY be a set of rectangular coordinate axes, the origin O being taken
at the left corner of the building Fig. 1.1. If x and y are the coordinates of various
elements and Kx and Ky their stiff nesses in the two directions, the coordinates (Xr,Yr) of
the centre of rigidity or the point of rotation are computed as
Xr = ΣKyx ……… 1
ΣKy
Yr = ΣKxy ……… 2
ΣKx
The rotational stiffness Ip of the structure about the centre of rotation Cr is
given by
Ip = Σ(KxY2 + KyX2) …… 3
If the torsional moment T= Ved …… 4
Where ed = 1.5e, the torsion shears Vx and Vy on any column line be computed as:
Vx = T. Y. Kxx……… 5
Ip
Vy = T. X. Kyy…… 6
Ip
Where Kxx, Kyy are the total stiff nesses of the column line under consideration and X and
Y are coordinates w.r.t the centre of rigidity Cr.
-4-
5. Y
4
3@7.5 3
=22.5m Xr
Cr †
Cm†
Yr 2
1
X
A B C D E
4@7.5m=30m
Fig: 1.1 Plan of an Asymmetric Building
-5-
6. CHAPTER 2
LITERATURE SURVEY
2.1 GENERAL
It has been observed repeatedly in strong earthquakes that the presence of
asymmetry in the plan of a structure makes it more vulnerable to seismic damages. There
are reports of extensive damages to buildings that are attributed to excessive torsion
responses caused by asymmetry in earthquakes such as the 1972 Managua earthquake
(Pomares Calero5 1995), the 1985 Michanocan earthquake (Esteva6 1987) and the 1989
Loma Prieta earthquake (Mitchell et al7 (1990)). Fig. 2.1 shows damages in a multi-storey
building after the 1995 Hyogoken-Nanbu earthquake in Kobe, probably caused by
excessive torsion responses because its core was eccentrically located in plan.
Asymmetry in plan causes torsion in a building because the centre of mass and the
centre of rigidity do not coincide. The distance between the two centers is termed
structural eccentricity and the magnitude of this eccentricity can be estimated. Torsion
can also arise in a building due to other sources for which estimating their magnitude is
difficult. Some examples of these sources for the so-called accidental torsion are the
rotational components in the ground motion, an unfavorable distribution of live load, and
the difference between computed and actual stiffness/mass/yield strength of the elements.
All these factors cause coupling between the lateral and torsion motions in a building that
leads to non-uniform distribution of in-plan floor displacement. This results in uneven
demands on the lateral resisting elements at different locations of the system.
-6-
7. Although torsion has long been recognized as a major reason for poor seismic
performance of multi-story buildings and many studies have been done on the seismic
torsion responses of single story buildings, the analytical and experimental studies on the
inelastic seismic response of multi-story buildings do not have a long history. The reason
as explained by De la Llera & Chopra8 (1995c) is that "most researchers have been
discouraged to look into the multi-story case in light of the already complex response of
single storey asymmetric buildings".
In most of the available studies on the seismic torsion response of multi-storey
buildings, simple building models such as shear walls are used and the conclusions of the
studies are based on the responses of buildings subjected to a limited number of
earthquake ground motions. Currently, there is no general agreement on how the torsion
effect should be allowed for in seismic design. These observations provided the
motivation for the study by A.S. Moghadam9 in order to provide a better understanding
of the problem of seismic damages caused by torsion in multi-storey reinforced concrete
frame buildings. Those investigations on torsion response that involve using the recorded
data in buildings during earthquakes are explained. Then experimental research is carried
out, and finally analytical work on the subject is explored.
2.2 STUDY ON RESPONSE OF BUILDINGS RECODRED IN EARTHQUAKES
Conducting experiments to study the inelastic response of a structure is not easy. To
obtain realistic estimations of the inelastic response, the test should be performed on a full-
scale prototype building. This is not practical for most structures. However, the recorded
motions of some instrumented buildings in earthquakes can provide valuable information
about the seismic performance of such buildings. Safak and Celebi10 (1990) introduced a
method to identify torsion vibration in an instrumented building. According to them,
similar methods can be used to identify inelastic behavior in vibrating structures. Lu and
-7-
8. Hall11 (1992) studied the data from two low-rise, extensively instrumented buildings in the
1987 Whittier Narrows Earthquake. Their study involved the investigation of responses of
buildings, responding in the elastic and marginally inelastic range, by comparing the
behavior of the buildings with computer simulations. Both buildings were modeled as
frame structures using a shear wall idealization. The recorded data at the basements were
used as the ground motion input for the models. The results from unidirectional ground
motion input were found to provide a reasonably close match of the actual responses during
the earthquake. Using bi-directional ground motion inputs gave an even better match to the
measurements. Sedarat et al.12 (1994) studied the torsion response characteristics of three
regular buildings in California, by analyzing the strong motions recorded in these
buildings during three recent earthquakes: the 1989 Loma Prieta earthquake, the 1986 Mt.
Lewis earthquake, and the 1984 Morgan Hill earthquake. The responses of the buildings
were compared with responses of models designed using the provisions of the 1988 UBC.
The results of their investigation indicated that the code provision was not adequate to
account for the torsion responses of these buildings.
2.3 EXPERIMENTAL STUDIES
Some experiments on scaled models are reported in the literature. Bourahla and
Blakeborough13 (1994) examined the performance of knee braces in asymmetric frame
buildings by designing and testing a one-twelfth-scale building model using a shaking
table. The test structure was a four-storey frame, three bays deep and three bays wide.
Several symmetric and asymmetric arrangements of the frame were tested. The changes
in responses due to asymmetry and also due to the unbalanced strength were investigated.
It was found that the effect of the unbalanced strength in a nominally symmetric frame
buildings is less significant compared with other sources of asymmetry. The energy
dissipation capacities of the frames were also studied. Based on the experimental results,
-8-
9. it is concluded that the magnitude of the eccentricity in itself is meaningless, but it is the
ability of the structure to resist torsion, which is critical.
2.4 ANALYTICAL STUDIES
Effects of torsion
Analytical studies have been done to compare the effects of torsion on the elastic
and inelastic behavior of buildings. Study of a seven-storey frame-wall structure (Sedarat
and Bertero 1990a14, 1990b15) demonstrated that linear dynamic analysis might
significantly underestimate the effect of torsion on the inelastic dynamic response of the
structure. On the other hand, the study of a thirteen storey regular space frame structure
Boroschek and Mahin16 (1992) showed that the effects of torsion were more severe if the
building is modeled as an elastic structure instead of an inelastic one, and the results were
found to be highly dependent on the characteristics of the earthquake motions. Therefore,
the issue of severity of torsion effect on the inelastic response of buildings has not been
settled.
Teramoto et al.17 (1992) presented some results of dynamic analyses of an
asymmetric 10-storey shear beam building. They used one earthquake record as the input
motion. A conclusion of this study is that mass eccentric and stiffness eccentric systems
behave differently. When mass eccentricity exists at upper floors only, the eccentricity
will also have some effects on the lower floors. However, stiffness eccentricity only
affects the floors where eccentricity exists.
Cruz and Cominetti18 (1992) used a five storey-building model in their study and
concluded that the overall ductility and the fundamental period of the building are the
parameters that most strongly affect the responses of the building.
In a study by De la Llera and Chopra19 (1996) they concluded that increasing the
torsion capacity of the building by introducing resisting planes in the orthogonal
-9-
10. direction, and modifying the stiffness and strength distribution to localise yielding in
selected resisting planes, are the two most important corrective measures for asymmetric
buildings.
2.5 DESIGN PROCEDURES
Several issues related to the design of multi-storey buildings and evaluation of
building codes have been studied in the literature. Bertero20 (1992) developed formulae
with the objective of considering the elastic and inelastic torsion in the preliminary design
of tall buildings. Bertero21 (1995) used the classical theorems of plastic analysis to
estimate the reduction in the strength of a special class of buildings. De la Llera &
Chopra22 (1995a) proposed a procedure for including the effects of accidental torsion in
the seismic design of buildings. Ozaki et al.23 (1988) proposed a seismic design method
for multi-storey asymmetric buildings. Azuhata and Ozaki24 (1992) proposed a method
for safety evaluation of shear-type asymmetric multi-storey buildings. In both of these
studies, the damage potential due to torsion is evaluated based on the shear and torsion
strength capacity and the design shear force and torsion moment for each storey of the
building.
In a study by Duan and Chandler25 (1993) on an asymmetric multi-storey frame
building model, they concluded that application of the static torsion provisions of some
building codes may lead to non-conservative estimates of the peak ductility demand,
particularly for structures with large stiffness eccentricity. In another study they
(Chandler and Duan25 1993) proposed a modified approach for improving the
effectiveness of the static procedure for regular asymmetric multi-storey frame buildings.
2.6 SHORTCOMINGS OF THE PREVIOUS ANALYTICAL STUDIES
The number of parameters required to mathematically define the elastic and
inelastic properties of a representative model of an asymmetric multi-storey building is
- 10 -
11. enormous. Therefore, all studies that have been reported in the literature involved using
simple models for the building and the conclusions are drawn based on a limited number
of earthquake records as ground motions input.
In almost all these studies, the multi-storey frame buildings are modelled as shear
buildings. The shear building model is not a good representative of the frame buildings in
a seismic zone because a shear building model has strong beams, which causes the plastic
hinges to occur at the columns. This is in contradiction to the strong column-weak beam
philosophy in earthquake design (Tso26 1994). A study by Moghadam and Tso27 (1996b)
has shown that shear-building modeling may lead to unreliable estimates of the important
design parameters. Rutenberg and De Stefano28 (1997) have pointed out that some of the
difference between the results of modeling a building as a shear building versus a ductile
moment resisting frame building in the study by Moghadam and Tso27 (1996b) might be
due to differences in the periods of the two compared models. Modeling of a building as a
shear building involves changing the stiffness of beams to very high values. This in turn
causes the period of the shear beam model to change. Therefore; modeling a ductile frame
building as a shear building will cause changes in not only the mode of failure, but also the
natural periods of the building. Thus, the relevance of observations of studies using shear
beam modeling to actual ductile moment resisting frame structures in seismic active
regions is questionable.
2.7 SIMPLIFIED METHODS
Some simplified approaches have been developed in the literature to estimate the
inelastic seismic responses of multi-storey buildings. De la Llera and Chopra8 (1995c)
developed a simple model for analysis and design of multi-storey buildings. Each storey
of the building is represented by a single super-element in the simplified model. The use of
storey shear and storey torque interaction surface (Kan and Chopra29 (1981), Palazzo and
- 11 -
12. Fraternali30 (1988), De la Llera and Chopra31 (1995b)) is an important component of this
method. The storey shear and torque (SST) surface is basically the yield surface of the
storey due to the interaction between storey shear and torque. Each point inside the
surface represents a combination of storey shear and torque that the storey remains
elastic. On the other hand, each point on the surface represent a combination of shear and
torque that leads to the yielding of the storey. It is shown that the SST surfaces can be
used for single storey systems and multi-storey shear buildings. One major assumption
embedded in the method is that the stories of a multi-storey building are considered as
independent single storey systems. In other words, the floor diaphragms are assumed
rigid, both in-plane and out-of-plane. This assumption of out-of-plane rigid diaphragms
is equivalent to assuming rigid beams in the building. How realistic is such a model to
represent the behavior of ductile frame buildings in seismic regions is a subject that
requires further investigation.
In the performance based design codes and in the guidelines for retrofitting of
buildings, the use of different versions of a static inelastic response analysis procedure,
commonly known as pushover analysis, has been suggested as a valid tool to evaluate the
acceptability of any proposed design, or to assess the damage vulnerability of existing
buildings. Moghadam and Tso32 (1996a) extended the application of the pushover
analysis to asymmetrical buildings by using a 3-D inelastic program. Kilar and Fajfar33
(1997) developed a simple method to conduct pushover analysis for asymmetric buildings
by modeling the building as a collection of planar macro-elements. Another method
proposed by Tso and Moghadam34 (1997) incorporates the results of elastic dynamic
analyses of the building in the pushover procedure. A further simplification is achieved
by requiring only a two-dimensional inelastic analysis program to perform the pushover
analysis on asymmetrical multi-storey buildings (Tso and Moghadam34 1997, Moghadam
- 12 -
13. and Tso35 1998). Rutenberg and De Stefano28 (1997) conducted pushover analyses on a 7-
storey wall-frame building and found reasonable agreement between results of pushover
and inelastic dynamic analyses.
- 13 -
14. The eccentric
elevator core
r Collapse of this column due
to excessive displacement
demand initiated the
progressive collapse in the
building
Fig 2.1 Example of Structural collapse caused by torsion (Eccentric elevator core lead to
significant torsion deformation and the collapse of corner columns)
A department Store in Kobe, Japan after 1995 Earthquake
- 14 -
15. CHAPTER 3
STRUCTURAL MODEL, LOADINGS &
RESPONSE PARAMETERS OF INTEREST
3.1 INTRODUCTION
The study in this work is based on the analyses of a family of structural models
representing multi-story asymmetrical buildings. These models are subjected to both
critical and lateral loadings expected on buildings during an earthquake. A set of response
parameters is used to illustrate the effect of torsion in these buildings.
The purpose of this chapter is to present the basic assumptions and the tools utilized
in this work. The different building configurations are introduced first. Then the methods
and the loadings used in the analyses are discussed. Finally the chosen response
parameters are outlined. The material presented in this chapter prepares the background
information for the results to be presented in the subsequent chapters.
3.2 BUILDING CONFIGURATIONS
The basic structural model used throughout this a study is uniform nine-story building;
asymmetric with respect to both X and Y axis to demonstrate many of the features
expected from multi-story buildings subjected to seismic loading. The assumed plan of
building is shown in Fig. 3.1. It has an L-shape floor plan of dimensions 42.4 m by 53.0
m, and a uniform floor height of 4.2 m Fig. 3.2. The plan considered is asymmetric. For
convenience, the X-direction is referred to as the main direction and the Y-direction is
referred to as the transverse direction. To resist the lateral loads, there are 28 RC columns
supporting to flat slab. The flat slab is of thickness 0.25 m with column caps 3.6x3.6x0.5
m with (post tensioned) edge beams of size 0.6x0.5 m are provided through out the
building in all floors. All the columns are placed at strategic locations with spacing of
- 15 -
16. 10.6x10.6 m, having 5 bays in X direction & 4 bays in Y direction. The grids are marked
as 1 to 6 in X direction and A to E in Y direction as shown in Fig. 3.1. The Seismic
analysis is carried out as per the latest IS-1893-2002 code by the Response Spectrum
technique. The buildings are assumed to be located in zone-II, zone-V and located on
three types of soils (Hard, Medium; Soft soils). The Response quantities considered
includes axial forces, moments in X & Y directions, twisting moments, %steel, steel area
etc. for the columns; further both ordinary moment resisting frame (OMRF) and special
moment resisting frame (SMRF) are considered.
3.3 COMPUTER SOFTWARE STAAD.Pro 2006
The static and dynamic behavior of the multi-story asymmetric buildings in the
elastic range is the main focus of the study reported in this work. Therefore computer
program with the ability of performing 3-D elastic static and dynamic analysis was
necessary. The program STAAD.Pro-2006 has been chosen as the base computer
software in performing the analyses. To have a clear understanding of the analysis a study
has been carried out to evaluate this program by comparing its results with the responses
derived from the manual calculations.
3.4 BASIC ASSUMPTIONS IN MODELING
The following are the main modeling assumptions used in this study.
3.4.1 MODELING OF THE BUILDING
• Rigid slab: It is assumed that all the columns in the buildings are connected by
floor diaphragms that are rigid in their own plane. Therefore every floor has only
two translational and one rotational degrees of freedom. The in-plane
displacements of all the nodes on the floor are constrained by these degrees of
freedom. However, the nodes can have independent vertical displacements.
- 16 -
17. • Fixed base: The columns of buildings are assumed to be fixed at their base on
rigid foundation. No soil-structure interaction effect is considered in this
study.
• One directional earthquake input: Only one direction of response values are
applied at the junction of columns and floor diaphragms. Due to the fixed base
assumption, all supports are assumed to move in phase. No vertical translation is
applied to the buildings.
• Lumped mass at floor level: The mass and the mass rotational moments of inertia
of the buildings are assumed to be lumped at the floor levels.
3.4.2 MODELING OF THE FRAMES
There are different analytical models available to simulate structural frames. In
this study an edge beam element with flat slab having and a column element are used to
model the elements of the frames in the buildings.
- 17 -
19. CHAPTER 4
ANALYSIS & DESIGN OF ASYMMETRICAL MULTI-
STOREY BUILDINGS INCORPORATING TORSIONAL
PROVISIONS
4.1 INTRODUCTION
In a symmetric building, all the lateral load-resisting elements at different
locations in plan experience the same lateral displacement when subjected to
unidirectional forces. As a result, the force induced in each element is proportional to its
lateral stiffness. This observation leads to a guideline that calls for assigning the design
strength of the lateral load-resisting elements according to their stiffness. In an
asymmetric building, however, the location of a lateral load-resisting element affects the
share of load that it should resist because the loadings on the rigid floors of these
buildings are accompanied by torques caused by the structural eccentricity in the
building. The force induced in each element from the floor torques is proportional to its
contribution to the torsion stiffness of the building. The torque-induced force in an
element is called the torsion shear. The location of an element not only determines the
magnitude, but also the direction of the torsion shear. Depending on the direction of the
torque, the torsion shear should be added to or subtracted from the forces induced in that
element by the translational displacement of the floors.
To compensate the torsion effect on the performance of a building, different
approaches have been suggested to replace the rule of distribution of strength among the
elements proportional to their lateral stiffness. These approaches can collectively be referred to
as torsion provisions. The goal of this chapter is to evaluate the effectiveness of a few torsion
provisions to improve the seismic performance of asymmetric multistory buildings.
- 19 -
20. The first approach that is studied here is distribution of the strength based on static
equilibrium consideration. Then the static torsion provisions based on the Indian seismic
code (IS: 1893-2002) are studied. Finally, the application of response spectrum analysis to
proportion the design strength of the elements is considered.
4.2 TORSIONAL PROVISIONS
Torsion provisions are incorporated in most building codes to redistribute the
strength among elements to minimize the torsion effects. Codes usually divide the
buildings into regular and irregular buildings and consider that static torsion provisions will
be suitable for regular buildings. For irregular buildings, design based on dynamic analysis,
such as the response spectrum method, is suggested.
4.3 I. S. CODE DESIGN PROVISIONS FOR TORSION
The static torsion provisions require the application of static torsion moments to
be included in the determination of the design forces. The product of the lateral force and
the design eccentricity determines the value of the torsion moment. The design
eccentricity can be different from the structural eccentricity in a building. To protect the
elements on both side of the building, codes require two separate load cases to be
considered involving two design eccentricities. The magnitudes of the two design
eccentricities are derived from equations:
(ed)x = l.5 e + 0.1 b ------- (4.1) ;
(ed)z = 0.5 e - 0.l b------ (4.2) ;
where (ed)x and (ed)z are the two design eccentricities, e is the structural eccentricity and
“b” is the width of the building. To design the elements, the forces required for resisting
the torsion moments (torsion shears) should be combined with the shear from
translational loading.
- 20 -
21. 4.4 CLASSIFICATION OF ASYMMETRICAL BUILDING USING
FREE VIBRATION ANALYSIS
One procedure to classify a building is to carry out a free vibration analysis.1 The
nature of a mode can be identified using the modal mass information derived from the
free vibration analysis. The first two mode shapes of the buildings and also the effective
modal masses of the first 12 modes of the buildings are presented. The mode shapes of
the buildings are given in two formats. In one format, the displacements and rotations at
CM of the floors are given for each mode. In the second format, the lateral displacements
of the five frames are shown for each mode.
Based on structural dynamics, it can be shown that translation predominant modes
in general have larger modal masses than torsion predominant modes.1 In the figures, the
effective modal masses are shown in figure: against the natural periods of the building. It
can be seen that the first mode is translation predominant in X-direction of the building.
The first translation predominant mode is the second mode as can be seen by the large
modal masses associated with the second mode for Y-direction of the building. In the
case of third mode purely torsion predominant, where as in first and second modes also
very less torsion values will be appearing, but predominant case is translational.
A parameter defined here as effective modal moment of inertia provides a
quantitative way of identifying the contribution of different modes to the displacements of
edge 1 and edge 6 of a building.1 Depending on the sign of this parameter one can show
whether the effects of the rotational and translational components of a coupled mode are
additive or subtractive on each edge of the building. The effective modal moment of
inertia for the nth mode is defined as I*On (Chopra 1995, where this parameter is called
modal static response for base torque)1:
- 21 -
22. N
I*On = Σ r2 Γ n m j φj θn
J=1
This equation is developed for an asymmetric building with eccentricity in one
direction only, such that floor rotations are coupled with floor displacements in the In-
direction. In the equation, N= total number of floors, n= the mode number, r= mass
radius of gyration, m = mass of floor, φj θn= the rotational element on the jth floor in the
n-th vibration mode shape.
Γn is defined as:
N
Σ m j φj y n
J=1
Γn = -------------------------------------------------------- (4.3)
N N
J=1
Σm j φ² j y n + r²
J =1
Σ m j φ² j θ n
Where φj y n is the translational element on j th floor in the n th vibration mode. The
effective modal moment of inertia idea is based on the concept of modal expansion
(Chopra, 1995)1 that uses the effective modal mass and the effective modal moment of
inertia to expand the effective force vector of a structure.
4.5 TORSIONAL ANALYSIS OF AN L-SHAPED BUILDING
36 2
The calculations of torsion seismic shears as per I.S. Code is illustrated for the L-
shape building shown in Fig. 3.1
Imposed load floor 39 = 4kN/m²; Imposed load roof 39= 1.5 kN/m²
Grade of concrete M35 and density 37 = 25 kN/m³, E 37 = 29.580 kN/m²
Floor finishes 38 = 60mm of 20 kN/m³
- 22 -
23. Column drop/cap = 3600x3600x0.5 depth (0.2 flat slab)
Column size = 0.9x0.9 = 0.054675m4, Partitions load 38 = 1.25 kN/m²
∴ Total additional dead load on the slab = 1.25 + 1.2 = 2.45 kN/m²
Note: - There is a 200mm thick block (brick) work around the building.
Storey shears:-
(i) Total weight of slab in a storey
a) 0.2(31.8 x 53+10.6 x 31.8)25 = 10112.4 kN
b) 2.45(31.8 x 53+10.6 x 31.8) = 4955.08 kN
15067.5 kN
(ii) Total weight of column caps(18 numbers ) = 0.3(3.6 x 3.6 x 18 No’s) 25
= 1749.6 kN
(iii) Total weight of column in a storey (28 numbers) = 0.9 x 0.9 x 4.2 x 25 x 28
= 2381.4 kN
(iv) Total weight of walls in a storey (½ above & ½ below floor) @ 20 kN/m³
= (31.8+10.6+10.6+31.8+42.4+42.4) 0.2 x 4.2 x 20
= 2849.28 kN
(v) Live load (50% during earthquake for 4KN/m² class loading)
= (31.8 x 53+10.6 x 31.8)0.5 x 4 = 4044.96 kN
Total weight lumped @ each floor of the 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, roof (9th
floor).
W1 = W2 = W3 = W4 = W5 = W6 = W7 = W8
(15067.5 + 1749.6 +2381.4 + 2849.3 + 4044.96) = 26092.76kN
Total weight lumped @ roof =W9
{15067.5 + 1749.6 +0.5(2381.4 + 2849.3) +0} = 19432.45 kN
- 23 -
24. Theoretical Base Shear = Vb = (Z/2 x I/R x Sa/g) W
Time period (In shorter direction) T = 0.09H/√Ds =0.09 x 37.8/√42.4 = 0.522 sec
Time period (In longer direction) T = 0.09H/√Ds =0.09 x 37.8/√53 = 0.497 sec
In longer direction Sa/g = 2.5, in shorter direction = 2.5
∴ VB = (0.1/2 x 1/3 x 2.5) 228174.53 kN
= 9507.3 kN
Vertical storey shear distribution for whole building can be determined using the
equation:-
Qi= Vb x Wi hi2
Σ Wi hi2
Floor Wi in kN hi Wi hi2 Qi Vi in kN
9(roof) 19433 37.8 27766648 2169.8 ≅2169.5
8 26093 33.6 29457953 2302 4471.5
7 26093 29.4 22553745 1762.44 6233.9
6 26093 25.2 16570099 1295 7528.9
5 26093 21.0 11507013 899.2 8428.1
4 26093 16.8 7364483 575.5 9003.6
3 26093 12.6 4142525 323.7 9327.3
2 26093 8.4 1841122 144 9471.3
1 26093 4.2 460280 36 9507.30
Σ Wi hi2 = 121663873
CENTRE OF MASS IN X- DIRECTION:
The total height acting along each of column line 1-1 to 6-6 for storey 1, 2, 3, 4, 5, 6, 7, 8
& 9(roof) can be computed as below mentioned table:
WEIGHT CALCULATION IN X- DIRECTION
Colu Weigh Weight Weigh Weigh Live Total weight Live Total
mn t of of slab t of t of load in in 1 to 8 load weight in
line beams in kN colum walls kN floors in kN @ roof 9th roof in
in kN n in in kN in kN kN
kN
1-1 145.8 1255.60 340.2 534.24 337.80 2612.92 - 2275.84
2-2 291.6 2511.20 340.2 178.08 674.16 3995.24 - 3321.08
3-3 340.2 2929.70 425.30 534.24 786.52 5013.96 - 4227.44
4-4 388.8 3348.30 425.30 356.16 898.88 5417.44 - 4518.56
5-5 388.8 3348.30 425.30 356.16 898.88 5417.44 - 4518.56
6-6 194.4 1674.20 425.30 890.4 449.44 3633.74 - 3184.30
ΣW=26090.74 ΣW=22045.78
- 24 -
25. WEIGHT CALCULATION IN Y- DIRECTION
Column Weight Weight Weight Weight Live Total Live Total
line of of slab of of load in weight in load weight
beams in kN column walls kN 1 to 8 @ in 9th
in kN in kN in kN floors in roof roof in
kN in kN kN
A-A 145.8 1255.60 340.2 534.24 337.08 2612.92 - 2275.84
B-B 388.8 3348.33 510.3 534.24 896.76 5678.43 - 4781.67
C-C 486.0 4185.4 510.3 356.16 1123.6 6661.46 - 5537.86
D-D 486.0 4185.4 510.3 356.16 1123.6 6661.46 - 5537.86
E-E 243.0 2092.7 510.3 890.4 561.8 4298.20 - 3736.40
ΣW=25912.47 kN ΣW=21869.63kN
CENTRE OF MASS IN X- DIRECTION
Taking moment of the weights @ about line “1-1”
Cmx (for 1 to 8 floors) =
(2612.92x0+3995.24x10.6+5013.96x21.2+5417.44x31.8+5417.44x42.4+3633.74x53)
26090.74
∴ Cmx = 743207.764 = 28.49 m
26090.74
Cmx (for roof 9th floor) =
(2275.84x0+3321.08x10.6+4227.44x21.2+4518.56x31.8+4518.56x42.4+3184.30x53)
22045.78
∴ Cmx (@ roof) = 628870.228 = 28.53 m
22045.78
CENTRE OF MASS IN Y – DIRECTION
Taking moment of the weights @ about line “A-A”
Cmz = (2612.92 x 0+5678.43 x 10.6+6661.46 x 21.2+6661.46 x 31.8+4298.20 x 42.4)
25912.47
(1 to 8 floors)
∴ Cmz = 595492.42= 22.98 m
25912.47
- 25 -
26. Cmz= (2275.84 x 0+4781.67 x 10.6+5537.86 x 21.2+5537.86 x 31.8+3736.4 x 42.4)
21869.63
(@ roof)
∴ Cmz (@ roof) = 502615.64 = 22.98 m
21869.63
CENTRE OF RIGIDITY IN X – DIRECTION
Lateral stiffness of column k = 12EI
L3
For a square column (0.9x0.9 mts) (having) (using)
M35 grade of “E” value same and also “L” are constant; kx = ky = k
xr = Σ ky. X
Σ ky
= (4k x 0+4k x 10.6+5k x 21.2+5k x 31.8+5k x 42.4+5k x 53)
28k
∴ xr = 784.4k = 28.014 m
28k
CENTRE OF RIGIDITY IN Y– DIRECTION
Zr= Σ kx.y
Σ kx
= (4k x 0+6k x 10.6+6k x 21.2+6k x 31.8+6k x 42.4)
28k
∴ Zr = 636k = 22.714 m
28k
Eccentricity:-
For 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th floors
ex = | Cmx - xr| = | 28.48 – 28.014| = 0.466 m
ez = | Cmz - Zr| = | 22.981 – 22.714| = 0.267 m
For 9th (roof) storey
ex = | Cmx - xr | = | 28.53 – 28.014| = 0.516 m
- 26 -
27. ez = | Cmz - Zr | = | 22.983 – 22.714| = 0.269 m
TORSIONAL STIFFNESS
Ip = Σ (kX. Y 2 + kY. X 2 )
Σ kX.Y2
= k [4(22.714)2+6(22.714-10.6)2+6(22.714-21.2)2+6(22.714-31.8) 2+6(22.714-42.4) 2]
Σ kX. Y2 = k [2063.703+880.494+13.75+495.33+2325.23]
∴ Σ kX. Y2 = 5778.51k m4
Σ kY.X 2 = k [4(28.014)2+4(28.014-10.6)2+5(28.014-21.2)2+5(28.014-31.8)2
+5(28.014-42.4)2+5(28.014-53)2]
= k [3139.137+1212.99+232.153+71.669+1034.785+3121.500]
∴ Σ kY.X2 = 8812.234k m4
Ip = Σ (kX. Z2 + kY. X 2 )
Ip = (5778.51 + 8812.23)
= 14590.74k m4
ADDITIONAL MOMENTS DUE TO SESMIC FORCE IN X- DIRECTION
(b = 42.4 mts)
1st floor
T1a = Vx (1.5 ez +0.05b)
= 9507.3(1.5 x 0.267+ 0.05 x 42.4) T1a = 23963.15 kNm
T1b = Vx (ez - 0.05b)
= 9507.3 (0.267 - 0.05 x 42.4) T1b = - 17617 kNm
2nd floor
T2a = 9471.3(1.5 x 0.267+0.05 x 42.4 T2a = 23872.4 kNm
T2b = 9471.3 (- 1.853) = -17550.3 kNm
- 27 -