Seismic Response of Mid-Rise Wood- Frame Buildings on Podium
Hysteresis Model of Low-rise Steel-frame Building and Its ...
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Steel Structures 6 (2006) 327-336 www.kssc.or.kr
Hysteresis Model of Low-rise Steel-frame Building and
Its Seismic Performance
Hirofumi Aoki* and Katsutoshi Ikeda1
Department of Architecture, Yokohama National University, Japan1Office of Industry and Community Liaison, Yokohama National University, AEI Structures and Materials, Japan
Abstract
The aim of this paper is to study the seismic response characteristic of low-rise steel-frame buildings under different typeof earthquakes, using artificial seismic waves for response analysis. These buildings are converted into a single-degree-of-freedom system and are modeled from slip-type to normal-bilinear-type load-displacement hysteresis model. Seismic waves aregenerated by considering the instantaneous energy input by differing the type of ground acceleration time history, from longduration to near fault seismic motions. The result of response analysis shows the balance between seismic motions energy andbuilding income capacity energy based on the influence of seismic motion and hysteresis model types of buildings. The growthof total energy input caused by the deformation of the structure into plastic area is investigated as well as the growth of naturalperiod.
Keywords: Seismic Performance, Low-rise Constructions, Steel-frame prefabricate house, Hysteresis Model, Artificial SeismicWaves
1. Introduction
The norm of seismic design in Japan is work to classify
building response caused by earthquakes as input motion
factors and to show definitive measures on ensuring the
function and the safety of structures. However, the seismic
time history response of the property and the scale of the
earthquake, as opposed to the force-displacement hysteresis
of buildings, shows an extremely disarranged result.
To this, Akiyama proposed a “Design method using the
balance between seismic motion and building income
capacity energy” and finally succeeded in clearing the
complicated relationship mentioned above (Akiyama, 1985).
This design method was legislated on June 2005 as “Structure
Design Method of Earthquake-resistant Calculation Based
on Energy Balance” in Japan.
However, even if the incoming seismic energy is
considered as a definite volume, the characteristic of seismic
motion input appears on the response of buildings. In The
Great Hanshin Earthquake of January 17, 1995, more
than one million of buildings collapsed due to a strong
near fault earthquake. The motion occurred directly above
its epicenter showing a strong impact, that is, an instantaneous
energy input into the building.
History of energy input, showing the impact degree of
seismic motion, is the points to be considered to prevent
structure collapse before the end of total energy input. On
a previous paper of Steel-housing structure (Ikeda, 2003),
using wooden panels fastened by screws to 1mm thick
steel studs frame, the analytical result shows the influence
of different seismic motion type with the same energy
spectrum input. Here, the displacement response increases
as the instantaneous energy input grows, where the seismic
motion type becomes a near fault motion.
To clarify this influence on different type of low-rise
steel frame building, seismic analysis is carried out using
slip to bilinear type load-displacement hysteresis model.
These structures are modeled by changing the energy
input capacity of its slip area.
Artificial seismic waves, from long duration to near
fault motions, are considered and used to define the seismic
performance of these buildings. These artificial waves are
generated using the method of Osaki (Osaki,1978) and by
considering the instantaneous energy input value
proposed by Kuwamura (Kuwamura, 1997).
Seismic performance of low-rise steel frame buildings
are represented in this paper, based on the method of
Akiyama by using energy response. The total input energy
of response is divided into several parts, to analyze the
influence of seismic motion type and also the hysteresis
model types of buildings.
*Corresponding authorTel: +81-45-755-1287; Fax: +81-45-759-1878E-mail: [email protected]
328 Hirofumi Aoki and Katsutoshi Ikeda
2. Modeling of Low-rise Steel-frame Building
2.1. Conversion of low-rise steel-frame building into
analysis model
In Japan, the number of housing built per year is about
1,200,000, and of these, prefabricate housing of less than
3 stories account for roughly 180,000. Prefabricate houses
are classified into steel-frame, wood-frame and concrete
structure, with steel-frame structure holding 70%, a number
of 130,000 (JPCSMA, 2004).
Conversion of low-rise steel-frame building into structure
model is examined in the study made by Yanai et al.
(2002). The research defines the scales of prefabricate
houses to put them in simple structure models for response
analysis by planning real scale low-rise buildings.
Plans are made by referring The Allowable Stress
Intensity Design Method of the Architectural Norm and
by selecting authorized design rules of prefabricated low-
rise steel buildings in Japan. The shape of planning is
specified and story stiffness (the floor is supposed to be
rigid) is then determined.
The scale of buildings is set up from 2 to 3 stories; the
limits of floor area and the ratio of each story floor area
are specified by referring statistical information (MIAC,
SB, 1998) to assume sufficient type of buildings. From
the statistical data, the total floor area is 80~280 (m2)
having an average of 126 (m2). The mass of the building
is then calculated using each floor surface.
Using the story stiffness and the mass of the building, the
first mode natural period and the yield shear force coefficient
of the 1st story for each building is finally given.
Table 1 shows the range and the average of the calculated
natural period and the yield shear force coefficient of the
1st story values of 3 stories building. These average
values are adopted here for seismic response analysis.
2.2. Hysteresis model
The construction of low-rise steel-frame prefabricated
houses is classified into 4 types: panel-type, frame-panel-
type (including frame-brace-type), moment frame type,
and unit type (BCJ, 1995).
Generally a non-buckling frame structure including
tension embrace have a slip-type hysteresis model (Fig.
1a), and a moment frame structure has a normal-bilinear-
type hysteresis model (Fig. 1b).
In slip-type hysteresis model, the displacement advances
without resistance in the slip area, but in some structure
this slip area can have consumption of energy for structural
reasons. For example, this behavior appears on steel-
housing structure in which steel-frame components are
fastened to wooden panel with tapping screws. In this
paper, the level of this slip area is expressed by using a
“semi-slip coefficient”, b, and with its use, “semi-slip-
type model (Fig. 1c)” can be modeled. The hysteresis
model is classified as slip-type when b = 0.0, and becomes
normal-bilinear-type when b approaches 1.0.
The ratio of elastic rigidity ke to plastic area rigidity kp
is shown as the “plastic rigidity ratio”, γ. This ratio values
are fixed by referring to authorize design rules of prefabricated
low-rise steel buildings.
When the hysteresis model is normal bilinear type, γ =
0.15 and γ = 0.015 for slip-type. To simplify the γ value
for semi-slip-type hysteresis model, the coefficient b is
determined by making a linear approach from γ = 0.015
to γ = 0.15.
γ = 0.135b + 0.015 (1)
where
γ = kp/ke
ke: elastic rigidity,
kp : plastic rigidity
b = 2δb/(δb = δy): semi-slip coefficient (2)
δy: the elastic limit deformation
δb: the Bauschinger deformation (refer Fig. 1(b))
2.3. Energy input into a mass system-skeleton-part of
plastic strain energy, Bauschinger-part of plastic
strain energy
The equation of motion for a single-mass oscillatory
system is expressed in the following Eq. (3):
Figure 1.
Table 1. Scale of buildings (3 stories building)
Range Average
Natural period T 0.40~0.50 0.45
Yield shear force coef. of 1st story α1 0.23~0.30 0.25
Hysteresis Model of Low-rise Steel-frame Building and Its Seismic Performance 329
(3)
where
m: mass
y: displacement of mass relative to the ground
c: damping constant
F(y): restoring force
y0: horizontal ground displacement
Multiplied by dy = dt on both sides, and integrated
over the entire duration of the earthquake, t0, Eq.(3) is
reduced to:
(4)
The right-hand side of the above equation expresses the
total amount of energy exerted by an earthquake. The
second term of left-hand side expresses the energy
consumed by the damping mechanism, Eh. The first term
of left-hand side expresses the kinetic energy at the
instant when the earthquake motion vanishes. The third
term expresses the strain energy stored in the spring
system, which consists of cumulative plastic strain energy,
Ep, and elastic strain energy at the instant when the
earthquake motion fades away. The kinetic energy and
the elastic strain energy constitute the elastic vibrational
energy, Ee. Therefore, Eq. (4) becomes as follows:
Ep + Ee + Eh = E (5)
In this paper, the cumulative plastic strain energy, Ep, is
divided into two parts, skeleton-part of plastic strain energy,
Eps, and Bauschinger-part of plastic strain energy, Epb.
Ep = Epb + Eps (6)
When the inexperienced displacement of the hysteresis
cycle is defined as skeleton-part, and the experienced
range displacement as Bauschinger-part (referring Fig. 2
(a)~(c)), the skeleton-part contributes directly to the
damage of the building, whereas the Bauschinger-part is
related to the law cycle fatigue damage. This relationship
can be shown by using the hysteresis model as given in
Fig. 2 (a) to (c).
When the damage limit of construction is expressed by
the extent of deformation, the cumulative plastic ductility
ratio, η, which is the division between elastic limit
deformation γy and the maximum plastic deformation
δpmax, is generally used. This is a characteristic value that
can directly be related to the skeleton-part of plastic strain
energy, Eps.
3. Characteristic of Input Earthquake
3.1. Generation of artificial earthquake
By analyzing the time history of ground motion
acceleration in the frequency domain, the earthquake
information is given by Fourier amplitude spectrum and
Fourier phase spectrum (Osaki, 1978).
In this paper, equivalent energy velocity response, VE,
established on notice (MLIT, 2005) (Fig-3) is used as
Fourier amplitude spectrum and represents the target
spectrum. Here, when m is the mass, VE is shown as
follows:
(7)
The typical value of standard deviation of Fourier phase
difference (Osaki, 1994) is also adopted in this paper, and
artificial earthquake is generated using Fourier inverse
transformation.
The shape of an earthquake envelope curve can be
controlled by Fourier phase difference spectrum, and
considering this as a normal distribution, by changing its
standard deviation of phase difference, σ = 0.06 × 2π~
0.21 × 2π at intervals of 0.03 × 2π, ground acceleration
time history of earthquakes from long duration to near
fault seismic motions can be generated (Kuwamura, 1997).
The normal distribution is defined by its average, µ,
and its standard deviation, σ and is expressed as follows
in, Eq. (8). The control of the average, µ, and the standard
deviation, σ on Fourier phase difference spectrum in
appropriated to the control of peak time and expanse on
the acceleration time history respectively.
(8)
my··
cy··
F y( ) my··0–=+ +
y·
1
2---my
·2cy·2
td0
t0
∫ F y( )y· td0
t0
∫+ + my··0y·td
0
t0
∫–=
VE
2E
m------=
ζ l( )1
2πσ------------- 0.5–
1 n µ–⁄σ
---------------⎝ ⎠⎛ ⎞exp=
Figure 2. Skeleton-part, Eps and Bauschinger-part of energy,Epb.
Figure 3. Equivalent energy velocity response spectrum, VE.
330 Hirofumi Aoki and Katsutoshi Ikeda
ζ(l): envelope curve function
n: the number of envelope curve
l: the division number of earthquake duration
µ: average of envelope curve
σ: standard deviation of envelope curve
Here, we assume that the average of the envelope
curve, µ, is 0.50, and that the seismic duration is 80 (sec),
to generate seismic motions having the same peak time in
their duration.
Figure 4 presents samples of generated 6 types × 5
waves, and is compared with its target equivalent energy
velocity response spectrum in Fig. 3. Having the same
energy velocity response spectrum, the peak of ground
acceleration increases as the phase difference, σ decreases,
where the earthquake becomes a near fault motion.
Figure 5 shows the comparison with total energy input
E-T. The generated waves conform well to the target
energy spectrum.
Figure 6 and Fig. 7 show the acceleration response
spectrum and the velocity response spectrum of generated
waves; respectively. Figure 8 and Fig. 9 show the maximum
value of, velocity response spectrum, (SV)max and the
acceleration response spectrum, (SA)max, respectively in
relation with σ/2π values.
This method as well as the legislated “Structure Design
Method of Earthquake-resistant Calculation Based on
Energy Balance” (MLIT, 2005) is based on a fixed
amount of total energy, and as a consequence, it is
important to note that the maximum response acceleration
of near fault seismic motion (σ/2π is smaller) becomes
bigger than that of long duration seismic motion.
Figure 10 shows the comparison of σ/2π between the
generated waves and four types of real earthquakes: The
Imperial Valley Earthquake El Centro NS record (1940),
Kern Country Earthquake Taft EW record (1952), The
Off Tokachi Earthquake Hachinohe EW record (1968),
The Great Hanshin Earthquake JMA Kobe NS record
(1995). We can see here from the σ/2π level of The Great
Hanshin Earthquake that σ = 0.06 × 2π represents an
extremely strong near fault seismic motion.
On a study of Steel-housing structure (Ikeda, 2003),
generated seismic waves are used for seismic analysis
referring to the Architectural Law of Japan. These waves
show an extremely long duration motion characteristic,
having a σ/2π of about 0.21 × 2π.
Figure 4. Samples of generated acceleration wave.
Figure 5. Total energy input spectrum, E of generated waves.
Figure 6. Velocity response spectrum, Sv of generated waves.
Figure 7. Acceleration response spectrum, Sa of generated
waves.
Hysteresis Model of Low-rise Steel-frame Building and Its Seismic Performance 331
4. Response Analysis
4.1. Hysteresis of restoring force, time history of
response deformation and energy input
Low-rise steel-frame building of 2 to 3 stories is directly
modeled into a single-degree-of-freedom system, for
seismic analysis. The model parameters shown in Fig. 11
are set referring real building mentioned in paragraph 2.1.
The coefficient of viscous dumping is given proportional
to stiffness.
Fig. 12 shows response analysis results, using typical
value of standard deviation of Fourier phase difference,
σ/2π = 0.06, 0.12, 0.21, and the semi-slip coefficient
b = 0.0, 0.5, 1.0 as parameters.
Starting from the top and moving down in Fig. 12, the
graphs represent hysteresis of the restoring force (Q)-
deformation (δ) relationship, time history of deformation
(δ), and time history of the total energy input E(t), plastic
strain energy Ep(t) and skeleton-part of plastic strain
energy Eps(t), respectively.
As the hysteresis model of building becomes slip-type
(b = 0.0), the rigidity in the plastic area and the Bauschinger-
part of plastic strain energy input, Epb, decreases. Therefore,
compared to bilinear type one (b = 1.0), the deformation
advances larger in the plastic area.
When the standard deviation of Fourier phase difference,
σ, that corresponds to the energy input ratio of the earthquake,
becomes bigger from near fault (σ = 0.06 × 2π) to long
duration (σ = 0.21 × 2π) seismic motions, the response
amplitude or displacement decreases and the total energy
input increases. This corresponds to the growth of the
maximum response acceleration with the decrease of
standard deviation of Fourier phase difference as shown
in (SA)max-σ/2π relationship (Fig. 9). At the same time, the
deformation advances in the plastic area, the response
deformation increases with the growth of the hysteresis
curve, and energy input E becomes bigger.
4.2. Total energy input
4.2.1. (b)-(VE, Vp, Vps) and (σ)-(VE, Vp, Vps) relation
Fig. 13 shows (b)-(VE, Vp, Vps), and (σ)-(VE, Vp, Vps)
relationships. As already shown in Fig. 12, the equivalent
energy velocity, VE, becomes smaller as the semi-slip
coefficient, b, increases and get closer to normal bilinear
model.
When the earthquake is a near fault motion having σ =
0.06 × 2π, the equivalent energy velocity, VE, increases
linearly from about 150 (cm/sec) to almost 250 (cm/sec)
as the semi-slip coefficient, b, decreases. This means that
about 1.5 times of equivalent energy velocity differentiates
here as the structure of the building changes from normal
bilinear to slip model.
The influence of seismic motion type appears especially
on normal bilinear model structures. As the structure
becomes a slip model, the equivalent energy velocity, VE,
stay stable with a maximum of 250 (cm/sec) whatever the
type of seismic motion differs.
4.2.2. Growth of energy input
The subject of this study is to analyze time history
response of prefabricate steel-frame houses of less than 3
stories. Its fundamental natural period of elastic system T
is set to 0.45 (sec). The target energy velocity response
spectrum of input earthquake used in this study is shown
in Fig. 14, and is set on the second ground level specified
in the Japanese Architectural Standard Law. The total
energy input when the fundamental natural period of
elastic system T = 0.45 should be iET=0.45= 6,845 (kN · cm)
where the equivalent energy velocity, VE= 117 (cm/sec),
however the result of energy input response analysis is
greater.
Figure 8. (Sv)max − σ/2π.
Figure 9. (Sa)max − σ/2π.
Figure 10. Real earthquakes.
Figure 11. Response analysis model.
332 Hirofumi Aoki and Katsutoshi Ikeda
To visualize this growth of energy, the ratio of total energy
input, E, to the total energy input when the fundamental
natural period of elastic system T = 0.45, iET=0.45, E/iET=0.45
is shown in Fig.15 (a) and (b).
Figure 15 (a) shows the difference between hysteresis
models and the ratio of energy input E/iET=0.45. The response
analysis result on normal-bilinear-type hysteresis model
(b = 1.0) is close to iET=0.45 value when the input seismic
motion is a long duration type having σ = 0.21, but as the
earthquake becomes a near fault motion type and the
semi-slip coefficient b approaches 0.0, the result of the
amount of energy input becomes near the upper limit of
the target energy velocity response spectrum iET=0.96 =
30,752 (kN · cm) (Fig. 14) where the equivalent energy
velocity, VE = 248 (cm/sec).
Here, when the earthquake is a near fault motion (σ =
Figure 12. Hysteresis of restoring force, time history of response deformation and energy input.
Hysteresis Model of Low-rise Steel-frame Building and Its Seismic Performance 333
0.06 × 2π), E/iET=0.45 is nearly 2.5 for normal bilinear model
structure, and take a bigger value than other seismic
motions.
Therefore, it is clear that the near fault seismic motion
of σ = 0.06 × 2π is giving the most critical result compared
to all earthquakes presented in this paper. For slip model
structure, the input energy to the building is 4 times bigger
than when the fundamental natural period of elastic
system is set to T = 0.45.
Fig. 14 (b) shows similar properties, and when the
hysteresis model is slip-type, the energy input approaches
the upper limit iET=0.96 and when the hysteresis model is
normal-bilinear-type, the energy input becomes nearby
iET=0.45 value.
One of the reasons for the above behavior is the growth
of natural period of the system caused by the deformation
into the plastic area.
4.2.3. Growth of natural period
Figure 15 (a) and (b) shows VE/VET=0.45 – Teq/T0.45
relationship. Here, VET=0.45 is the equivalent energy velocity
when the fundamental natural period of elastic system T
= 0.45, that is VET=0.45 = 117 (cm/sec), Teq is the estimated
natural period of the system after the deformation in
plastic area, using Fig. 14 applied from the total energy
input E, T0.45 is the natural period of elastic system,
T = 0.45.
Similar to Fig. 15, the deformation advances considerably
in the plastic area and the natural period extends, especially
when the hysteresis model is slip-type in Fig. 16 (a), and
when seismic motion is near fault in Fig. 16 (b).
Here, the growth of natural period becomes about 4.5
times for slip model structure and a maximum of 3.5
times for normal bilinear structure.
Taking notice on long duration seismic motion shown
in Fig. 16 (b), The growth of equivalent energy velocity
is from 1.0 to 1.5, however this value increases to a
maximum of 2.0, as the earthquake becomes a near fault
motion.
As a result, the importance of considering earthquake
as near fault motion is shown for the growth of total
energy input as well as the grown of natural period
occurred by deformation advance in the plastic area of
hysteresis model.
4.3. Partition of total energy input
As shown in paragraph 2.3., the cumulative plastic
strain energy, Ep, is divided into two parts, the skeleton-
Figure 13. (b)-(VE, Vp, Vps) and (σ)-(VE, Vp, Vps) relation.
Figure 14. Energy velocity spectrum and natural period ofbuilding.
334 Hirofumi Aoki and Katsutoshi Ikeda
part of plastic strain energy, Eps, and the Bauschinger-part
of plastic strain energy, Epb in this research.
Fig. 17 shows the ratio of the plastic strain energy, Ep,
and the skeleton-part of plastic strain energy, Eps, to total
energy input. Here, the skeleton-part plastic strain energy,
Eps, contributes directly to the damage of building.
As already shown in Fig. 13, the equivalent energy
velocity becomes smaller as the standard deviation of
Fourier phase difference, σ, increases. However, the ratio
of Eh and Ep, to the total energy input E and the ratios of
Eps, Epb to Ep, differ on the value of b, and it is interesting
that the ratio of Eh to Ep gives the relative maximum
nearby b = 0.4~0.5, on semi-slip-type hysteresis model
building. On the other hand, the skeleton-part of plastic
strain energy input Eps decrease as b approaches 1.0
(normal-bilinear-type hysteresis model building).
When the structure is semi-slip type having b = 0.5, the
type of seismic motion does not influence the ratio of Ep
to total energy input E which takes about Ep/E = 0.7.
However, for a normal bilinear model, as the earthquake
becomes a long duration motion, Ep/E decreases from 0.7
to 0.5.
Referring to the calculation of plastic strain energy, Ep
of Akiyama (Akiyama, 1985), using dumping factor, h,
Ep = E × 1/(1 + 3h + 1.2 )2 (5)
when h = 0.05, the calculation of Ep = 0.497E. This result
fits with the analysis result only when the earthquake is
a long duration motion.
Compared to the ratio of plastic strain energy, Ep/E, the
ratio of skeleton-part of plastic strain energy, Eps/E shows
more stability. When the seismic motion is near fault, that
is, the most critical type of earthquake (as explained on
paragraph 4.2.2.), the ratio, Eps/E stabilizes on a ratio of
about 0.5 for any type of structure.
Using this ratio, Eps/E, the total energy input can be
estimated by calculating or measuring the skeleton-part of
plastic strain energy, Eps, of the structure. the skeleton-
part of plastic strain energy, Eps, can be also estimated
from the total amount of input energy, to determine the
damage degree of the structure.
However, to improve the accuracy of this method of
damage estimation, further analysis must to be carried
out, changing for example the level of total energy input
and the modeling of low-rise steel-frame structure.
5. Conclusions and Recommendation for Structure Design
Time history response analysis is carried out on low-
rise steel-frame prefabricated houses of less than 3
stories. The fundamental natural period of elastic system
of these buildings is in a short-range period and is fixed
to T = 0.45 (sec). The time history response analyses are
h
Figure 15. Growth of total energy input.
Figure 16. Growth of natural period and equivalent energyvelocity.
Hysteresis Model of Low-rise Steel-frame Building and Its Seismic Performance 335
executed using artificial earthquake having the same
target energy spectrum based on “Structure Design Method
of Earthquake-resistant Calculation Based on Energy
Balance” of Japan (MLIT, 2005). These earthquakes are
classified from long duration to near fault seismic motions,
and are generated by changing the instantaneous energy
input with the use of standard deviation of Fourier phase
difference. The building force-displacement is modeled
from slip-type hysteresis model such as braced steel-
frame building to normal-bilinear-type model such as
moment frame building. To precise the damage of the
structure, the cumulative plastic strain energy, Ep, of total
energy input, E, is analyzed in dividing it into two parts,
skeleton-part of plastic strain energy, Eps, and Bauschinger-
part of plastic strain energy, Epb.
Following the balance between seismic motions and
building income capacity energy, the seismic performance
characteristic of these construction models is given
below;
(1) The total energy input grows with the deformation
of the structure into the plastic area, in the case the
earthquake is a near fault motion witch has the highest
instantaneous energy input; the input energy to the
building is 4 times bigger than when the fundamental
natural period of elastic system is set to T = 0.45. Even if
the earthquake is a long duration motion, the input energy
ratio decreases, but takes a value of nearly 2.5 for normal
bilinear model structure.
(2) With the deformation progress of structure in the
plastic area, the natural period grows and extends from
short period to longer period; when the seismic motion is
a near fault, the growth of natural period becomes about
4.5 times for the slip model structure and 3.5 times for the
normal bilinear structure in maximum.
(3) The ratio of cumulative plastic strain energy Ep, to
the total energy input E differs on the form of the
structure as well as the seismic motion type, however, the
ratio of skeleton-part of plastic strain energy, Eps/E,
stabilize on a value of about 0.5 for each type of structure,
when the seismic motion is near fault. The value can be
used for the estimation of total input energy or the
damage degree of low-rise steel-frame structure.
As this analysis is carried out with a single-degree-of-
freedom system, the result cannot predict damage like
concentration in specific story of building. Further more
analysis is necessary to approve seismic behavior of low-
rise steel-frame buildings.
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