Hysteresis Model of Low-rise Steel-frame Building and Its

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.


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.


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.


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.


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.


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.


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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Figure 17. (b)-(Ep/E, Eps/E) and (σ/2π)-(Ep/E, Eps/E) relation.


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Hysteresis Model of Low-rise Steel-frame Building and Its