INTRODUCTION
A compendium of field observations and analytical results indicate that
certain failure modes are more convincingly attributable to high vertical
earthquakeinduced forces. In addition to the possibility of compressive
overstressing or failure due to direct tension, vertical motion may induce
failure in shear and flexure. Under reduced compression or mild tension,
the contribution of concrete to shear resistance is eroded, thus many
observed shear failure modes may have underlying vertical motion effects
(Papazoglou and Elnashai, 1996).
The presence of the vertical excitation can produce a variation in the
distribution of the dissipated energy among the elements of the frames,
with a possible greater demand in the columns (Decaninni et al.,
2002).
The main effect of the vertical motion consists of the variation of axial
force in the columns. The high values of compression, or even tension,
induced by the vertical excitation could produce damage in the structure
which leads to a decrease of structural capacity to withstand the horizontal
seismic motion, resulting in an increase of horizontal displacements (Papazoglou
and Elnashai, 1996; Decaninni et al., 2002; Diotallevi and Landi,
2000; Kikuchi and Yoshimura, 1984; Hosseini and Nezamabadi, 2004; Wang
et al., 1989).
The varying axial force in the columns results in pinched hysteretic
behavior that causes larger horizontal displacement and column end moments
and curvature (Diotallevi and Landi, 2006; Foutch, 1997).
The vertical component has the important effect of changing the plastic
hinge distribution, sequence of hinging and mode of failure of the structure
(Diotallevi and Landi, 2006; Ghobarah and Elnashai, 1998). Vertical ground
acceleration in nearsource regions can increase the design forces for
connections of heavy nonstructural cladding panels (Memari et al.,
2004).
The responses subjected to vertical acceleration (V) are usually taken
into consideration by using about two thirds of the horizontal response
spectra (H). However, recent studies have shown that the V/H response
spectral ratio depends on the distance of the site to the seismic source.
The ratio is higher in the nearfield region and in the highfrequency
range of the response spectra. The V/H ratio largely exceeds the commonly
assumed ratio of two thirds at short periods in the nearfield regions.
As a result, the margin of safety of structures subjected to an earthquake
in the nearfield regions is questionable (Kianoush and Chen, 2006; Xinle
et al., 2007).
Assessment of structural performances during past earthquakes demonstrates
that plan irregularity, due to asymmetric distribution of mass, stiffness
and strength, is one of the most frequent sources of severe damage, since
it results in floor rotations (torsional response) in addition to floor
translations. In past years, large research efforts have been devoted
to the study of the seismic response of asymmetric structures, both in
the elastic and inelastic range of behavior (Rutenberg, 1998). In particular,
inelastic behavior is of great interest since the ability of structures
to resist strong earthquakes depends upon their ductility and capacity
for energy dissipation.
Most of these studies were conducted by using simple single storey asymmetric
models. Simplified models neglect important effects that may influence
inelastic behavior of resisting elements and in turn, of the entire structure.
Namely, resisting elements are assumed to resist unidirectional horizontal
forces only; therefore, no allowance for interaction among bidirectional
horizontal and vertical forces in resisting elements is usually made (Stefano
and Pintucchi, 2002).
One of the past investigations that have considered the effect of vertical
component of earthquake motion acting simultaneously with horizontal component
on the torsionally coupled structures has been done by Gupta and Hutchinson
(1994). They have evaluated the displacement response of a simple lumped
mass model of a single story building resting on a rigid foundation with
three degree of freedom: firstly, the lateral displacement (u), secondly,
the torsional displacement, u_{θ} and thirdly, the vertical
displacement, w. The mass of diaphragm was acting at an eccentricity (e)
from the center of resistance (Fig. 1a, b).
The mass of vertical elements (columns) of building was lumped in line
with the center of resistance and the mass of the diaphragm in z direction
was ignored to avoid creating eccentricity in the vertical direction.
In this study, the model has been improved by adding one degree of freedom,
rotation about xaxis and considering eccentricity (e) of vertical masses.
In response analysis, it has been assumed that the earthquake ground
motion input in horizontal and vertical direction ,
are applied uniformly over the base of structure. The displacement responses
subjected to horizontal motion (u), the torsional motion (u_{θ}),
the combined horizontal and torsional (u+u_{θ}), the vertical
motion (v), the rotational motion about x axes (v_{θ}) and
finally the combined vertical and rotational (v+v_{θ}) have
been evaluated in terms of time history displacement for the ElCentro
earthquake, NS, 1940.
STRUCTURAL SYSTEM
A single story building, shown in Fig. 1a, c
consists of a rigid floor diaphragm of radius r supported on elastic column
has been considered. The four degrees of freedom of the system are: the
lateral displacement (u) of the floor relative to the ground along the
principal axis of resistance (x) of the building; the torsional displacement
(u_{θ}) of the floor about the vertical axis (z); the vertical
displacement (v) of supporting columns, relative to the ground along the
principal axis of resistance (z) of the building and rotational displacement
(v_{θ}) of the floor about the horizontal axis (x).
The total mass of the floor is represented by m. The structural eccentricity
(e), between the center of mass and resistance is caused by different
mass densities ()
in the two halves of the equivalent floor disc, split along y = 0, as
shown in Fig. 1a. The principal axis of resistance
coincide with the x and y horizontal axis of the reference system. The
principal axis of resistance in the vertical direction coincides with
the z axis of the reference system. Rotational displacements (θ)
of the disc about z and xaxis take place about the center of resistance
(x = y = 0).
The response displacements of the floor diaphragm have been assumed to
be sufficiently small to ensure the structural stability and therefore,
the effects of changes in the geometry may be neglected. The damping ratio
(ζ) is taken as 5% of critical damping.
EQUATION OF MOTION
The Equations of motion of the building model (Fig. 1a,
c) are derived from the dynamic equilibrium for each
degree of freedom.
xdirection:
Rotation about z direction (θ_{z}):
zdirection:

Fig. 1: 
Lumped mass model of a single storey building; (a,
b) the model developed by Gupta and Hutchinson (1994) and (a, c)
the model improved in this study 
Rotation about x direction (θ_{x}):
Substituting;
and writing Eq. 14 in matrix form we obtain:
where,
are mass, damping, stiffness and ground motion matrices, respectively.
Evaluation of damping matrix coefficients
Considering classical damping:
where, a and b are arbitrary proportionality factors and are determined
by assuming that both the natural modes of vibration (u and u_{θ})
of the coupled system have the same ratio ζ of critical damping.
The damping matrix proportional to the mass and/or stiffness matrices
will permit uncoupling the Equation of motion. For each mode the generalized
damping is given by:
where, ω_{i} is the natural frequency of the coupled system
in mode i and ζ is the modal damping ratio.
Solving above Equations yields:
where, ω_{1} and ω_{2} are the fundamental
and secondary natural frequencies of the coupled system.
Substituting a and b into Eq. 6 yields:
Comparing Eq. 13 and 5:
where, ω_{u}, ω_{θz}, ω_{v},
ω_{θx} represent the translational, torsional, vertical
and rotational natural frequencies, respectively, of the corresponding
torsionally uncoupled building (zero eccentricity):
By defining:
from Eq. 1418 damping matrix coefficients are obtained
as function of frequency ratio value.
By substituting damping matrix coefficient (Eq. 2327)
into damping matrix, it is obtained:
NATURAL FREQUENCIES
The undamped natural frequencies of the system are determined from a
free vibration analysis as the following:
Since:
Equation 29 becomes:
The frequency determinant may be written by simplifying.
TIME HISTORY ANALYSIS
For comparing the results of this study with those of the study developed
by Gupta and Hutchinson (1994), ElCentro record has been used in time
history analysis. For the purpose of computing the vertical displacement
response subjected to ElCentro earthquake, 100% of the NS has been assumed
to be acting in the vertical direction. The peak displacement of this
record is 20.91 cm, the peak velocity is 32.48 cm sec^{1} and
the peak acceleration is 0.312 g.
Three structural models have been prepared and analyzed: first, the new
model, presented in this study, second, the model presented by Gupta and
Hutchinson (1994), third the model presented by Gupta and Hutchinson (1994)
in which there are 6 degrees of freedom and distributed vertical springs
in perimeter of slab. These three models have been analyzed for three
values of frequency ratios (λ = λ_{Tx} = λ_{v}
= λ_{Tz} = 0.6, 1.0 and 1.4) and two values of eccentricity
ratios (e_{r} = 0.15 and 0.30) separately. In order to develop
displacement response spectra, for each eccentricity ratio value and each
frequency ratio value, analysis have been repeated for ten values of uncoupled
natural period (T_{u} = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6,
1.8, 2.0). At first, first and second models have been analyzed by using
MATLAB software and then three models have been analyzed by using SAP2000.
In analysis it has been assumed that the earth ground acceleration input
in horizontal and vertical direction is applied uniformly over the base
of the structure. The structure have been modeled to be resting on a rigid
foundation, hence interaction effects have been neglected.
TIME HISTORY ANALYSIS RESULTS
In the present study the individual response quantities have been computed
separately as a complete time history; the response maxima u_{max},
u_{θmax}, v_{max } and v_{θmax} have
been then selected as appropriate. These maximum displacement responses
are presented in normalized form in Fig. 27. In each
figure, response curves have been normalized to the maximum quantity in
the earthquake ground motion records and plotted as a function of the
uncoupled natural period (T_{u}). As it is expected the horizontal
displacement response curves of center of resistance and edges are quite
close to each other for all frequency ratio values and all eccentricity
ratio values. For example, horizontal displacement response curves of
center of resistance and edges for λ_{Tz} = 1.0 and e_{r}
= 0.30 are shown in Fig. 2.
The vertical displacements of edges in each time step are obtained as
(v+v_{θ}) and (vv_{θ}) where v is the vertical
displacement of center of resistance and v_{θ} is rotational
displacement about xaxis. The normalized vertical displacement curves
are shown in Fig. 35. Paying attention to these figures,
one can observe that:
Large frequency ratio value (λ = 1.4)
• 
The vertical displacement curves of center of resistance are quite
close to each other for e_{r} = 0.15 in entire range of T_{u}
= 0 to 1.0 sec 
• 
The vertical response curve of center of resistance of second model
is upper than the other models for e_{r} = 0.30 while the
vertical response curve of edges of this model is lower than the others
because of rotation resulting from mass eccentricity 
• 
The edges displacement response of second and third models are close
to each other for e_{r} = 0.15 while the response curve corresponding
to first model exceeds the other two response curves for e_{r}
= 0.30 

Fig. 2: 
Normalized horizontal displacement response spectra
(λ_{Tz} = 1.0, e_{r} = 0.30); (a) translational
displacement of center of resistance and (b) maximum translational
displacement 

Fig. 3: 
Normalized vertical displacement response spectra (λ_{Tz}
= λ_{Tx} = λ_{v} = 1.4); (a, c) vertical
displacement of center of resistance and (b, d) maximum vertical displacement
of edges 
• 
The edges responses increase up to 24% when eccentricity ratio value
is increased (Fig. 3) 
Intermediate frequency ratio value (λ = 1.0)
• 
The vertical displacement curves of center of resistance are close
to each other for e_{r} = 0.15 when uncoupled natural period
is less than 0.8 sec and more than 1.2 sec 

Fig. 4: 
Normalized vertical displacement response spectra (λ_{Tz}
= λ_{Tx} = λ_{v} = 1.0); (a, c) vertical
displacement of center of resistance and (b, d) maximum vertical displacement
of edges 
• 
The edges displacement response trends of first and third models
are about the same for e_{r} = 0.15 while the response curve
corresponding to first model exceeds the other two response curves
for e_{r} = 0.30 
• 
The edges responses increase up to 21% when eccentricity ratio value
is increased (Fig. 4) 
Small frequency ratio value (λ = 0.6)
• 
The vertical displacement curves of center of resistance are close
to each other for e_{r} = 0.15 in entire range of T_{u}
= 0 to 1.2 sec and for e_{r} = 0.30 when uncoupled natural
period is less than 1.0 sec 
• 
The edges displacement response trends of first and third models
are about the same for e_{r} = 0.15 while the response curve
corresponding to first model exceeds the other two response curves
for e_{r} = 0.30 when uncoupled natural period is less than
1.0 sec 
• 
The edges responses increase up to 14% when eccentricity ratio value
is increased (Fig. 5) 
By comparing the results of various frequency ratio values it is found
that the response differences decrease with decreasing frequency ratio
values (λ) for all eccentricity ratio values. In fact the increasing
rate of vertical response due to eccentricity is reduced by decreasing
torsional and rotational rigidity.
Normalized translational displacement curves corresponding of first model
are shown in Fig. 6 for various frequency ratio values.
The response curves are close together when uncoupled natural period is
less than 0.8 sec and then the curves corresponding to λ = 0.6 exceeds
the other two response curves up to T_{u} = 1.6 sec.
The higher response is obtained for the curve λ = 1.0 when T_{u}≥1.6
sec. The maximum translational displacement of edges is about 0.6 for
all frequency ratio values and for all eccentricity ratio values when
T_{u} = 0.8 sec.
Normalized vertical displacement curves corresponding of first model
are shown in Fig. 7 for various frequency ratio values.
The response curve corresponding to λ = 0.6 exceeds the other two
response curves for λ = 1.0 and λ = 1.4 over the entire range
of uncoupled natural period values. There is a sharp increase when T_{u}≥1.2
sec. This result is similar to result presented in the study of Gupta
and Hutchinson (1994).
The response values are the same when T_{u} = 0.8 sec. for λ
= 1.0 and λ = 1.4. When T_{u} = 1.0 sec the response values
are the same for λ = 0.6 and λ = 1.0 except for the maximum
vertical displacement response for e_{r} = 0.30. The vertical
displacement response increases significantly when the eccentricity ratio
value is increased.

Fig. 5: 
Normalized vertical displacement response spectra (λ_{Tz}
= λ_{Tx} = λ_{v} = 0.6); (a, c) vertical
displacement of center of resistance and (b, d) maximum vertical displacement
of edges 

Fig. 6: 
Normalized translational displacement response spectra
for new model; (a, c) translational displacement of center of resistance
and (b, d) maximum translational displacement 

Fig. 7: 
Normalized vertical displacement response spectra for
new model; (a, c) vertical displacement of center of resistance and
(b, d) maximum vertical displacement of edges 
CONCLUSION
In this study, a simple lumpedmass model of a single story building
resting on a rigid foundation with four degrees of freedom has been developed.
The mass of diaphragm (m) has been assumed to act at an eccentricity (e)
from the center of resistance (first model). Second model is model presented
by Gupta and Hutchinson (1994) and third model is the model of Gupta and
Hutchinson (1994) with 6 degrees of freedom and distributed vertical springs
in perimeter of slab.
Results of first model have been compared with results of the other two
models. Analysis have been carried out for three values of frequency ratio
and for two values of eccentricity. In order to develop displacement response
spectra, for each eccentricity ratio value and each frequency ratio value,
analysis have been repeated for ten values of uncoupled natural period.
The following results can be obtained:
• 
The horizontal displacement response curves of center of resistance
and edges are quite close to each other for all frequency ratio values
and all eccentricity ratio values 
• 
The vertical displacement response of center of resistance are close
together when uncoupled natural period is smaller than 1.0 sec, while
the vertical displacement responses of edges corresponding to proposed
model exceeds the other two model in this interval 
• 
The influence of higher eccentricity ratio value (e_{r})
is to increase the translational and vertical responses for smaller
frequency ratio values (λ) 
• 
The torsional and rotational responses increase significantly when
the eccentricity ratio value (e_{r}) is increased 
• 
The vertical response is very sensitive to the frequency ratio values
(λ). For smaller frequency ratio values, the response is much
higher when uncoupled natural period is greater than 1.2 sec 
• 
Increasing rate of vertical response due to eccentricity ratio value
is reduced with decreasing frequency ratio values (λ). In fact
reduction of tortional and rotational rigidity in comparison with
translational rigidity results in reduction of response differences 
• 
The responses obtained from proposed model are more than those of
the other two models in most cases and are increased when frequency
ratio values (λ) decrease 
Base of the above results it can be concluded that the proposed model
leads to more conservative response value in the period range of zero
to one second which is the period range of short to mid story building.
Further studies are required for the case of eccentricities in 2 direction
of x and y.
Notation:
a, b 
= 
Rayleigh`s damping constants 
C_{uu}, C_{uθ}, C_{θθz},C_{vv},
C_{vθ}, C_{θθx} 
= 
Coefficients of damping matrix 
C 
= 
Damping matrix 
C_{i} 
= 
Generalized damping in mode i 
D_{uu}, D_{uθ}, D_{θθz},D_{vv},
D_{vθ}, D_{θθx} 
= 
Dimensionless damping coefficient 
e 
= 
Structural eccentricity 
e_{r} 
= 
Eccentricity ratio value (e/r) 
g 
= 
Gravity acceleration 
k_{u}, k_{θz}, k_{v}, k_{θx} 
= 
Lateral, tortional, vertical and rotational story stiffness, respectively 
M 
= 
Mass matrix 
K 
= 
Stiffness matrix 
r 
= 
Radius of equivalent floor disk 
t 
= 
Time 
T_{u} 
= 
Uncoupled natural period 
u, u_{θ}, v, v_{θ} 
= 
Time history of translational, torsional, vertical and rotational
displacement of floor, respectively 

= 
Time history of displacement, velocity and acceleration in lateral
direction of building 

= 
Time history of torsional displacement, torsional velocity and torsional
acceleration of building 

= 
Time history of displacement, velocity and acceleration
in vertical direction of building 

= 
Time history of rotational displacement, rotational velocity and
rotational acceleration of building 
x, y, z 
= 
Cartesian axes of reference 
ζ 
= 
Viscous damping ratio 

= 
Tosional, vertical and rotational frequency ratio values, respectively 
ω 
= 
Excitation circular frequency 
ω_{n} 
= 
Natural frequency of torsionally coupled system 

= 
Translational, torsional, vertical and rotational natural frequencies,
respectively 
Ω_{n} 
= 
Coupled natural frequency ratio (ω_{n}/ω_{u}) 

= 
Mass densities of two halves of equivalent floor disc 