NISTIR 6903

Distribution of Earthquake Input 

Energy in Structures 

Payam Khashaee 

Bijan Mohraz 

Fahim Sadek 

H.S. Lew

John L. Gross 

United States Department of Commerce 

Technology Administration

National Institute of Standards and Technology 
NISTIR 6903

Distribution of Earthquake Input 

Energy in Structures 

Payam Khashaee 

Bijan Mohraz 

Fahim Sadek 

H.S. Lew 

John L. Gross 

Building and Fire Research Laboratory 

National Institute of Standards and Technology 

Gaithersburg, MD 20899 

January 2003 

U.S. Department of Commerce

Donald L. Evans, Secretary

Technology Administration 

Philip J. Bond, Under Secretary of Commerce for Technology 

National Institute of Standards and Technology 

Arden L. Bement, Jr., Director

ABSTRACT 

In developing an energy-based design approach and assessing the damage potential of 

structures, one must know the distribution of earthquake input energy among energy 

components: kinetic, elastic strain, hysteretic, and damping. This report examines the 

influences of the ground motion characteristics: intensity, frequency content, and duration 

of strong motion and the structural properties: ductility, damping, and hysteretic behavior

on the distribution of input energy for a one- and a five-story building using 20 

accelerograms, ten with short and ten with long duration of strong motion. 

Results indicate that for certain damping ratios, ductility has a significant influence on 

input energy and its distribution among energy components in a structure. For a given 

ductility ratio, small damping ratio (less than 5%) has a minor effect on input energy, but 

a major influence on the energy distribution. Damping ratios larger than 5% have a 

significant influence on the input energy and its distribution. Three energy ratios that 

relate to hysteretic energy were computed: the maximum ratio of hysteretic to input 

energy (Eh/Eir)m, the ratio of the maximum hysteretic energy to the maximum input 

energy Ehm/Eirm, and the equivalent number of yield excursions Neq=Ehm/(Fy.up) where Fy

is the yield strength, and up is the plastic deformation. It is found that (Eh/Eir)m generally 

reflects the energy demand for the largest yield excursion, and Ehm/Eirm and Neq reflect the 

energy demand for the entire duration of accelerogram. The study shows that (Eh/Eir)m is 

independent of the duration of strong motion and period of structure; however, Ehm/Eirm is 

independent of both only for periods less than 1 s. Results indicate that as the duration

becomes longer the equivalent number of yield excursions Neq increases indicating more 

structural damage. 

The influence of ground motion characteristics and structural properties on the 

distribution of energy parameters for a five-story building with fixed-base, base-isolation, 

supplemental damping, and semi-active control are examined using the 20 accelerograms. 

The results show that: 1) the distribution of energy through the height of the building is 

mostly independent of the frequency content and the duration of strong motion, 2) baseisolation, supplemental damping, and semi-active control reduce the damage potential by 

reducing the input and hysteretic energy demands and have significant influences on the 

distribution of energy through the height of the building. 

iii

ACKNOWLEDGEMENT 

This study was supported by the Structures Division, Building and Fire Research 

Laboratory, National Institute of Standards and Technology, U.S. Department of 

Commerce through a contract with Southern Methodist University. 

v

TABLE OF CONTENTS 

ABSTRACT ..........................................................................................................................iii 

ACKNOWLEDGEMENT ....................................................................................................v 

TABLE OF CONTENTS.......................................................................................................vii 

LIST OF TABLES.................................................................................................................ix 

LIST OF FIGURES ...............................................................................................................xi

1. INTRODUCTION ....................................................................................................1 

2. SUMMARY OF PREVIOUS RESEARCH ..............................................................3 

3. EARTHQUAKE INPUT ENERGY .........................................................................5 

3.1 Energy Balance Equation .............................................................................5 

3.2 Energy Ratio Time-histories..........................................................................6 

4. ESTIMATING THE EARTHQUAKE INPUT ENERGY .......................................11 

4.1 Existing Procedures for Computing Input Energy.........................................11 

4.2 The Influence of Ground Motion Characteristics on Energy Spectra............14 

4.3 The Influence of Structural Properties on Energy Spectra ............................20 

5. HYSTERETIC ENERGY DEMAND .......................................................................31 

6. ENERGY RATIOS IN AN MDOF STRUCTURE ..................................................37 

7. DISTRIBUTION OF INPUT ENERGY IN STRUCTURES WITH BASEISOLATION, SUPPLEMENTAL DAMPING AND SEMI-ACTIVE CONTROL .45 

7.1 Energy Equations for Buildings with Base-Isolation and Semi-Active 

Control ..........................................................................................................45 

7.2 Numerical Results .........................................................................................49 

8. CONCLUSIONS .......................................................................................................61 

8.1 SDOF structures ............................................................................................61 

8.2 A 5-story building with fixed base, base-isolation, supplemental damping, 

and semi-active control .................................................................................62 

REFERENCES .....................................................................................................................65 

vii
APPENDIX A. BOUC-WEN MODEL ......................................................................69 

APPENDIX B. EARTHQUAKE INPUT ENERGY FOR MDOF STRUCTURES...71 

APPENDIX C. LIST OF SYMBOLS..........................................................................73 

viii
LIST OF TABLES

Table 4.1 Strong motion records used in the analysis .................................... 15

Table 6.1 Inter-story ductility, normalized story shear, percentage of energy in 

each story, and the energy ratios (Eh/Eir)m, Ehm/Eirm, and Edm/Edm for the 

five-story building subjected to the S00E component of El Centro, the 

Imperial Valley, CA earthquake of May 18, 1940 scaled to an EPA = 

0.4g. ............................................................................... 39

Table 6.2 Influence of yield deformation on maximum inter-story ductility μmax, 

base shear Qb, and total energy parameters for the five-story building

subjected to the S00E component of El Centro, the Imperial Valley, CA 

earthquake of May 18, 1940 scaled to an EPA = 0.4g. .....................

 

41

Table 7.1 The average percentage of hysteretic energy, input energy, and story 

shear in each story for the five-story building with fixed-base, base 

isolation, supplemental damping, and semi-active control subjected to 

20 accelerograms scaled to an EPA = 0.4g. .................................... 

 

53

Table 7.2 (a) Maximum drift, maximum relative displacement, ductility ratio, and 

maximum story shear for each story, and (b) maximum damping, 

maximum hysteretic, maximum controller, and maximum input energies 

for the four buildings (with fixed-base, base-isolation, supplemental 

damping, and semi-active control) subjected to the S00E component of 

El Centro, the Imperial Valley, CA earthquake of May 18, 1940 scaled 

to an EPA = 0.4g. ................................................................ 55

Table 7.3 The energy ratios Ekr/Eir, Ed/Eir, Es/Eir, Eh/Eir, and Ec/Eir for the the four 

buildings (a) fixed-base, (b) base-isolated, (c) semi-active with small 

control forces, (d) and semi-active with large control forces subjected to 

the S00E component of El Centro, the Imperial Valley, CA earthquake 

of May 18, 1940 scaled to an EPA = 0.4g. ................................... 58

ix

LIST OF FIGURES 

Figure 3.1. Energy time-histories for a bilinear SDOF structure with Tn = 0.5 s, μ = 

3, α = 0.02, and ξ = 5% subjected to the S00E component of El Centro, 

the Imperial Valley, CA earthquake of May 18, 1940 ........................ 7 

Figure 3.2. Time-histories of the ratio of energy components to relative input energy 

for a bilinear SDOF structure with Tn = 0.5 s, a ductility ratio μ = 3, and 

5% damping subjected to the S00E component of El Centro, the 

Imperial Valley, CA earthquake of May 18, 1940. ............................ 8 

Figure 3.3. Time-histories of Eh/Ehm and Eh/Eir for a bilinear SDOF structure with Tn

= 0.5 s, a ductility ratio μ = 3, and 5% damping subjected to the S00E 

component of El Centro, the Imperial Valley, CA earthquake of May 18, 

1940. ................................................................................ 9 

Figure 4.1. Relative input energy for ductility ratio μ = 3 and a damping ratio ξ = 

5% for the S00E component of El Centro, the Imperial Valley, CA 

earthquake of May 18, 1940 scaled to EPA = 0.3g, 0.4g, and 0.6g. ........ 17

Figure 4.2. Ratio of the maximum hysteretic energy to the maximum input energy 

for ductility ratio μ = 3 and a damping ratio ξ = 5% for the S00E 

component of El Centro, the Imperial Valley, CA earthquake of May 18, 

1940 scaled to EPA = 0.3g, 0.4g, and 0.6g. .................................. 18

Figure 4.3. (a) Earthquake input energy and (b) hysteretic energy for the N65W 

component of Temblor, Park-field, CA earthquake of June 27, 1966 (tdi

= 6 s) and the S69E component of Taft Lincoln School Tunnel, Kern 

County, CA earthquake of July 21, 1952 (tdi = 29 s) for μ = 1, 3, and 5, 

and a damping ratio ξ = 5% scaled to an EPA = 0.4g. ........................ 19

Figure 4.4. Predominant periods Tpa and Tpe from the elastic acceleration and input 

energy spectra, respectively, for three accelerograms with short duration 

of strong motion and three accelerograms with long duration. ............ 21

Figure 4.5. Elastic input energy spectra for the 90o

 component of Sylmar County 

Hospital Parking Lot, Northridge, CA earthquake of January 17, 1994 

and the S16E component of Pacoima Dam, San Fernando, CA 

earthquake of March 22, 1971 scaled to an EPA = 0.4g ...................... 22

Figure 4.6. Relative input energy Eir for ductility ratios μ = 1, 2, 3, and 5, and a 

damping ratio ξ = 5% for three accelerograms with short, and three

accelerograms with long duration of strong motion .......................... 23

xi
Figure 4.7. Relative input energy Eir for a ductility ratio of 3, and damping ratios ξ = 

0, 2, 5, 10, 20, and 40% for three accelerograms with short duration of 

strong motion, and three accelerograms with long duration ................. 25

Figure 4.8. Energy ratio spectra for bilinear behavior with a ductility ratio μ = 3 and 

damping ratios 0%, 2%, 5%, 10%, 20%, and 40% subjected to the 360o

component of Sylmar County Hospital Parking Lot Station, Northridge 

earthquake of January 17, 1994. (a) Ekrm/Eirm, (b) Esm/Eirm, (c) Ehm/Eirm, 

(d) (Eh/Eir)m, (e) Edm/Eirm, and (f) Edism/Eirm. ................................... 26

Figure 4.9. Force-displacement diagram for an SDOF MRF and an SDOF EBF with 

Tn = 2.0 s, a ductility ratio of 3, and a 5% damping ratio subjected to the 

360o component of Sylmar County Hospital Parking Lot, Northridge, 

CA earthquake of January 17, 1994. ............................................. 27

Figure 4.10. Relative input energy Eir for a ductility ratio of 3, damping ratio ξ = 5%,

and two hysteretic behaviors for (a) the 360o

 component of Sylmar

County Hospital Parking Lot, Northridge, CA earthquake of January 17, 

1994, and (b) the S00E component of Mexico City earthquake of Sep 

19, 1985. .............................................................................. 29

Figure 5.1. (a, b) Maximum ratio of the hysteretic to input energy (Eh/Eir)m for 

records with short and long durations of strong motion, respectively. (c, 

d) Ratio of the maximum hysteretic energy to the maximum input 

energy Ehm/Eirm for records with short and long durations, respectively. 

(e, f) Equivalent number of yield excursions Neq for records with short 

and long durations, respectively for a ductility ratio μ = 3 and a 5% 

damping ratio. ........................................................................ 32

Figure 5.2. Time-histories of Eh/Ehm and Eh/Eir for a bilinear SDOF structure with Tn

= 1.0 s, a ductility ratio μ = 3, and 5% damping subjected to the 90o

component of Sylmar County Hospital Parking Lot, Northridge, CA 

earthquake of January 17, 1994. .................................................. 33

Figure 5.3. Time-histories of Eh/Ehm and Eh/Eir for a bilinear SDOF structure with Tn

= 2.0 s, a ductility ratio μ = 3, and 5% damping subjected to the 90o

component of Sylmar County Hospital Parking Lot, Northridge, CA 

earthquake of January 17, 1994. .................................................. 35

Figure 5.4. The mean (Eh/Eir)m and Ehm/Eirm spectra for bilinear behavior with 5% 

damping and ductility ratios μ = 2, 3, and 5 for 20 accelerograms. ........ 36

Figure 6.1. Total hysteretic and total input energies for the five-story building with 

uy1 = 0.5 cm subjected to the 20 accelerograms scaled to an EPA = 0.4g. 38

xii
Figure 6.2. (a) Relative displacement urel, (b) ductility demand μ, (c) normalized 

story shear, (d) Ehm/Eirm, (e) distribution of Eh though the height, and (f) 

distribution of Eir through the height for the 5-story building with yield 

deformations of the first story uy1 = 0.3, 0.5, 0.7, 1.0, and 1.5 subjected 

to S00E component of El Centro, the Imperial Valley, CA earthquake of 

May 18, 1940 scaled to an EPA = 0.4g. ........................................ 42

Figure 6.3. Distribution of energy through the height of a 5-story building with uy1 = 

0.5 cm subjected to 20 accelerograms scaled to an EPA = 0.4g. ............. 43

Figure 7.1 (a) Five-story building with fixed base, (b) with base-isolation, and (c) 

with supplemental damping or semi-active control subjected to the S00E 

component of El Centro, the Imperial Valley, CA earthquake of May 18, 

1940 scaled to an EPA = 0.4g...................................................... 46

Figure 7.2. Distribution of energy through the height of a 5-story building with base 

isolation (10% damping, Tb = 2.0 s, and uyb = 4.0 cm) with uy1 = 0.5 cm

subjected to 20 records scaled to an EPA = 0.4g. .............................. 50

Figure 7.3. Distribution of energy through the height of a 5-story building with 

supplemental damping (total damping 40%) with uy1 = 0.5 cm subjected 

to 20 records scaled to an EPA = 0.4g. ........................................ 51

Figure 7.4. Distribution of energy through the height of a 5-story building with 

semi-active control (r = 10-5 and q = 10+5) with uy1 = 0.5 cm subjected to 

20 records scaled to an EPA = 0.4g. ............................................. 52

Figure 7.5. Percentage of (a) hysteretic energy Eh, (b) input energy Eir, and (c) story 

shear Q distributed in the stories of the 5-story building with fixed base, 

base isolation, supplemental damping, and semi-active control subjected 

to 20 records scaled to an EPA = 0.4g. ........................................ 54

Figure 7.6. (a) Relative displacement, (b) ductility demand, (c) story shear, (d) 

Ehm/Eirm, (e) (Eh/Eir)m, and (f) Edm/Eirm for the five-story buildings: fixedbase, base-isolated, semi-active with large control force, and semi-active 

with small control force with uy1 = 0.5 cm and ξ1 = 5% subjected to the 

S00E component of El Centro, the Imperial Valley earthquake of May 

18, 1940 scaled to an EPA = 0.4g. ............................................... 57

Figure 7.7. Drift time-histories for the first story of the buildings with fixed-base, 

base-isolation, supplemental damping, and semi-active control subjected 

to the S00E component of El Centro, the Imperial Valley, CA

earthquake of May 18, 1940 scaled to an EPA = 0.4g. ....................... 59

xiii
Figure 7.8. Total input energy time-histories for the five-story buildings with fixedbase, base-isolation, supplemental damping, and semi-active control 

subjected to the S00E component of El Centro, the Imperial Valley, CA 

earthquake of May 18, 1940 scaled to an EPA = 0.4g ........................ 59

Figure A.1. Single degree of freedom system used for Bouc-Wen model ............... 70

xiv
1. INTRODUCTION 

The current seismic design practice which is based on strength principles (using the 

acceleration spectra) does not directly account for the influence of the duration of strong

motion nor for the hysteretic behavior of the structure. The hysteretic behavior is 

addressed indirectly by using the response modification factor R which is based primarily 

on the structural system selected. A design approach based on energy (Vision 2000), on 

the other hand, has the potential to address the effects of the duration and hysteretic 

behavior directly. 

The earthquake input energy transmitted to a structure consists of the kinetic energy, 

elastic strain energy, damping energy, and hysteretic energy. Kinetic energy reflects the 

work of the inertia force. Elastic strain energy is the portion of the input energy stored in 

the structure in the form of elastic strain. Damping energy is the work of the damping 

force. Hysteretic energy is the energy dissipated through the hysteretic action and is 

associated with the damage potential of the structure (Kuwamura and Galambos, 1989). 

Input energy reflects the intensity of earthquake ground motion. It is a more appropiate 

measure of the intensity than the peak ground acceleration or the effective peak 

acceleration defined by ATC 3-06 (1978) as 

EPA = Sa / 2.5 (1.1)

where Sa is the average spectral acceleration in the constant acceleration region, or the 

Spectrum Intensity defined by Housner (1975) as 

∫ = 2.5

0.1

SI PSV dTn (1.2)

where PSV is the pseudo spectral velocity and Tn denotes the period of structure in 

seconds. Earthquake input energy accounts for both the ground motion characteristics 

(frequency content, intensity, and duration of strong motion) and structural properties 

(ductility, damping, period, and hysteretic behavior), whereas PGA, EPA, and SI account 

for some but not all these characteristics. 

A design approach based on energy reflects the distribution of input energy and is suited 

for seismic design of structures with base isolation system and/or supplemental damping 

devices. In an energy-based design approach, once the energy demand for a structure is

estimated from the earthquake ground motion, the damage potential can be quantified by 

a combination of response and energy parameters (see Park and Ang, 1985). Sufficient

strength and energy dissipation capacity should be provided in the structure for an 

acceptable damage threshold, i.e. a desired performance objective. 

1
In seismic design of structures, the strength demand is defined by the story shear and the 

energy demand should be defined by the hysteretic energy for each story. While the

strength and energy dissipation capacities can be increased by using members with larger 

area and section modulus or by using materials with greater yield strength, the energy 

dissipation capacity can be increased by using ductile structural systems such as the 

eccentrically braced frames (EBF) instead of concentrically braced frames (CBF), see 

Newmark and Rosenbluth 1971. The energy dissipation capacity for moment resisting 

frames can be increased by providing special detailing in steel and reinforced concrete 

buildings (refer to IBC 2000). Computing the energy dissipation capacity of various 

structural members is not within the scope of this report (one should refer to Uang and 

Bertero, 1990). Only the energy demand is addressed herein. 

This report examines the influences of ground motion characteristics and structural 

properties on earthquake input energy and its distribution among energy components for 

a one- and a five-story building using 20 accelerograms, ten with short and ten with long

duration of strong motion. A summary of previous research on this subject is presented 

in Chapter 2. Chapter 3 presents the energy balance equation and the energy timehistories for an SDOF structure. Chapter 4 examines the existing procedures for 

computing the earthquake input energy, and how ground motion characteristics and 

structural properties influence the input energy. In Chapter 5, the relationship between 

the hysteretic energy and damage potential is examined. The distribution of story shear 

and energy in a 5-story building without control devices, with base isolation system, with 

supplemental damping devices, and with semi-active control are presented in Chapters 6 

and 7. Finally, the conclusions are discussed in Chapter 8. 

2
2. SUMMARY OF PREVIOUS RESEARCH 

McKevitte et al. (1980) computed the input energy, hysteretic energy, and the ratio of the 

cumulative hysteretic energy to input energy for SDOF and MDOF structures (three- and 

ten-story) with different structural properties subjected to four earthquake records (El 

Centro 1940, Taft 1952, Parkville 1956, and Pacoima Dam 1971). They concluded that 

the energy dissipated through inelastic deformation depends on the force-deformation 

characteristics, yield strength, and damping. They observed that the percentage of input

energy dissipated by the hysteretic action was approximately the same for different

records. McKevitte et al. concluded that the ratio of the maximum hysteretic energy to 

the maximum input energy for an MDOF structure can be estimated from an SDOF 

structure with the same fundamental period, yield strength, and damping. 

Zahrah and Hall (1984) computed the input energy for eight earthquake records and 

proposed an equivalent number of yield excursions to quantify the earthquake damage

potential. They observed that ductility, damping, and the post- to pre-yield stiffness 

ratios have small effects on the input and hysteretic energies for a structure with bilinear 

behavior. They stated that the equivalent number of yield excursions may be different for

different accelerograms. 

Akiyama (1985) compared the input energy from the Fourier Spectra of ground 

acceleration for a five-story building with different structural properties, and for an 

equivalent one-story building having the same fundamental period, total mass, and yield 

strength using the S00E component of the 1940 El Centro record. He showed that the 

total input energy transmitted to a five-story building can be computed from the input

energy transmitted to the equivalent one-story building with the same fundamental 

period, damping, and yield strength. 

Nakashima et al. (1996) investigated the effect of large post- to pre-yield stiffness ratios 

(up to 0.75) for bilinear SDOF and MDOF structures (5, 10, 13, 15, 17, and 20 story) 

subjected to three long duration records (El Centro 1940, Taft 1952, and Miyagiken-oki 

1978). They concluded that in general, large post- to pre-yield stiffness ratios have a 

minor effect on the input energy; however, as the post- to pre-yield stiffness ratio 

increases, the input energy spectra increase in the vicinity of the predominant period. 

They also concluded that the energy transmitted to an MDOF structure can be computed

from the input energy transmitted to an equivalent SDOF structure. 

Rahnama and Manuel (1996) computed the input energy, the hysteretic energy, and the 

ratio of the cumulative hysteretic energy to input energy for ductility ratios of 2 and 6 

with 5% damping for six sets of 19 accelerograms each: actual records, simulated records 

with the same duration as the actual records, and four sets of simulated records with 

durations of 5, 10, 15, and 20 s. They concluded that as the duration increases, the input 

and hysteretic energies increase. The duration, however, does not influence the ratio of 

cumulative hysteretic energy to input energy. 

3
Goel (1997) investigated the distribution of energy in asymmetric structures. He

concluded that the input energy transmitted to a symmetric or to an asymmetric one-story

building is approximately the same. He also showed that in an asymmetric building the 

hysteretic energy demand is larger for flexible elements, whereas it is approximately the 

same for stiff elements. 

4
3. EARTHQUAKE INPUT ENERGY 

Uang and Bertero (1990) proposed two procedures for computing the earthquake input 

energy: one based on the absolute motion and the other on relative motion. The 

difference between the two procedures is less important in damage assessment since the

hysteretic energy, which is associated with the damage potential of structures, is 

independent of the approach used. Chopra (1995), and Bruneau and Wang (1996)

believe that the input energy in terms of the relative motion is more meaningful than the 

input energy in terms of the absolute motion since internal forces within a structure are 

computed using relative displacements and velocities. Therefore, the procedure based on 

the relative displacement is used herein. 

3.1. Energy Balance Equation

The energy balance equation for an SDOF structure can be written as (Uang and Bertero

1990) 

∫ ∫ ∫ ∫ + + = − t

g

t

s

t t

m u u d c u d f u d m u u d

0 0 0

2

0

&& & τ & τ & τ && & τ (3.1) 

where m is mass of the structure, c is the damping coefficient, is the restoring force, u

is the relative displacement of the mass with respect to ground, is the velocity of the 

mass with respect to ground, is the acceleration of the mass with respect to ground,

 denotes the ground acceleration, and t is time. The energy balance equation (3.1) can

be written as

s f

u&

u&&

ug &&

Ekr + Ed + Ea = Eir (3.2)

where Ekr is the relative kinetic energy, Ed is the damping energy, Ea is the absorbed

energy, and Eir denotes the relative input energy such that 

2 E 1⁄2 mu kr = & (3.3)

∫ = t

Ed cu dt

0

2 & (3.4)

∫ = t

a s E f u d

0

& τ (3.5)

 (3.6) ∫ = − t

ir g E m u u d

0

&& & τ

5
The absorbed energy consists of the recoverable elastic strain energy Es and the 

irrecoverable hysteretic energy Eh where 

E f k s s / 2 2 = (3.7)

Eh = Ea − Es (3.8)

in which k is the pre-yield stiffness of the structure. 

Figure 3.1 shows the energy time-histories for a bilinear SDOF structure with a pre-yield

period Tn = 0.5 s, a ductility ratio μ = 3, a pre-yield damping ratio ξ = 5%, and a post- to

pre-yield stiffness ratio α = 0.02 subjected to the S00E component of El Centro, the 

Imperial Valley, California earthquake of May 18, 1940. The Bouc-Wen model (Bouc, 

1967; Wen, 1976) is used to characterize the nonlinear behavior of the structure, see 

Appendix A. Parameters used for characterizing the bilinear behavior with Bouc-Wen

model are: A = 1, β = γ = 0.5, n = 20, and α = 0.02. To overcome the problem associated 

with the non-zero initial conditions of the records, particularly in non-linear analyses (see 

Pecknold and Riddell, 1978), a one-second acceleration pulse is added to the beginning 

of the accelerogram. 

Figure 3.1 shows that the maximum kinetic and elastic strain energies occur in the initial 

stages of the excitation, whereas the maximum damping, hysteretic, and input energies 

occur at the end of the excitation. Therefore, the duration of strong motion significantly 

affects the maximum damping energy, the maximum hysteretic energy, and the 

maximum input energies, but not the maximum kinetic energy and the maximum elastic 

strain energy. 

3.2. Energy Ratio Time-Histories

Figure 3.2 shows the time-histories of the ratio of energy components to input energy for 

a bilinear SDOF structure with a pre-yield period Tn = 0.5 s, a ductility ratio μ = 3, and a 

5% pre-yield damping ratio subjected to the S00E component of the 1940 El Centro 

accelerogram. The figure indicates that the time-histories of the Ekr/Eir and Es/Eir ratios 

are similar and have large oscillations during the linear portion of the response. The peak

Ekr/Eir and Es/Eir ratios decay rapidly as the structure experiences nonlinear deformation 

because a significant portion of the input energy is distributed among hysteretic and 

damping energies rather than kinetic and elastic strain energies. The maximum Eh/Eir

ratio occurs during the largest yield excursion where approximately 45% of the maximum 

hysteretic energy Ehm is dissipated, see Figure 3.3. The hysteretic energy dissipated in the 

second yield excursion is 32% of Ehm. 

Figures 3.2 and 3.3 indicate that the maximum ratio of the hysteretic to input energy 

(Eh/Eir)m and the ratio of the maximum hysteretic energy to the maximum input energy 

Ehm/Eirm are different, i.e. (Eh/Eir)m=77% and Ehm/Eirm=55%. The (Eh/Eir)m ratio reflects

6
Figure 3.1. Energy time-histories for a bilinear SDOF structure with Tn = 0.5 s, μ = 3, α = 0.02, and 

ξ = 5% subjected to the S00E component of El Centro, the Imperial Valley, CA earthquake of May 

18, 1940

0

1

0 10 20 30 40

Time (s)

E

nergy (

m2/s2

)

Eir Eh Ed

Ekr Es

Damping

Hysteretic

Input

E krm = 0.07 (m

2

/s2)

E hm = 0.42 (m

2

/s2)

E sm = 0.02 (m

2

/s2)

E irm = 0.79 (m

2

/s2)

E dm = 0.36 (m

2

/s2)

50

7
Figure 3.2. Time-histories of the ratio of energy components to relative input energy for a bilinear 

SDOF structure with Tn = 0.5 s, a ductility ratio μ = 3, and 5% damping subjected to the S00E

component of El Centro, the Imperial Valley, CA earthquake of May 18, 1940. Parameters for BoucWen Model are A = 1, β = γ = 0.5, n = 20, α = 0.02

0

1

0 10 20 30 40 5

Time (s)

R

atio of energy com

ponents t

o i

nput ener

g

y

Ekr/Eir

Es/Eir

Eh/Eir

Ed/Eir

0

8
Figure 3.3. Time-histories of Eh/Ehm and Eh/Eir for a bilinear SDOF structure with Tn = 0.5 s, a

ductility ratio μ = 3, and 5% damping subjected to the S00E component of El Centro, the Imperial 

Valley, CA earthquake of May 18, 1940. Parameters for Bouc-Wen Model are A = 1, β = γ = 0.5, n = 

20, α = 0.02

0

1

0 10 20 30 40

Time (s)

Eh/Ehm and

Eh/Eir

Eh/Eir

Eh/Ehm

32% of E hm in the

second yield excursion

45% of E hm in the

first yield excursion

E hm = 0.42 (m

2

/s 2

)

E irm = 0.79 (m

2

/s 2

)

E hm /E irm = 0.55

50

9
the damage potential associated with the largest yield excursion, and the Ehm/Eirm ratio 

reflects the damage potential associated with the total number of yield excursions during 

the excitation. The differences of the two ratios are examined in Chapter 5. 

In the 1994 Northridge earthquake, more than 150 buildings with steel moment resisting 

frames experienced brittle fracture at the welded connections, and in some cases damage 

occurred during the first few seconds of the excitation (SAC 95-06). Damage

experienced in the initial stages of the excitation may have been caused by the 

insufficient energy dissipation capacity during the largest yield excursion (usually the 

first two yield excursions). Hence, in developing a damage index, both the damage

potential for the largest yield excursion and for the entire record should be addressed. 

Although proposing a damage index is not within the scope of this report (refer to Park 

and Ang, 1985), the relationship between the hysteretic energy and damage potential is 

discussed in Chapter 5. 

10
4. ESTIMATING THE EARTHQUAKE INPUT ENERGY 

In an energy-based seismic design, one needs to estimate the input energy in a structure 

and distribute it to various structural components. For an acceptable damage level, vision 

2000 (SEAOC 1995) and Park et al. (1987) recommend estimating the strength and 

hysteretic energy demands and providing sufficient strength and energy dissipation

capacity to withstand the design ground motion. 

This Chapter presents: 1) the existing methods for computing the earthquake input 

energy, 2) the influence of ground motion characteristics on the input energy, and 3) the 

influence of structural properties on input energy. 

4.1 Existing Procedures for Computing Input Energy

Housner (1956) computed the input energy per unit mass as 

2 1⁄2(PSV )

m

Ei = (4.1)

where m is the mass and PSV denotes the pseudo-spectral velocity. He used equation 

(4.1) for elastic and inelastic behavior. 

Zahrah and Hall (1984) computed the input energy per unit mass as 

 ∫ = − t

g

i u u d

m

E

0

&& & τ (4.2)

which is the same as equation (3.6). 

Akiyama (1985) proposed the input energy per unit mass for an elastic SDOF structure as 

2 1⁄2( ) E

i V

m

E = (4.3)

where is an equivalent velocity. He recommended the following values for V (in 

m/s): 

VE E

 VE = 2.5 Tn for Tn ≤ TG 

 VE = 2.5 TG for T ≥ T (4.4) n G

11
where is the predominant period of ground motion as a function of soil type. The 

values of T are 0.4, 0.6, 0.8, and 1.0 s for soil types I (bed rock), II, III, and IV (softest

soil), respectively. 

TG

G

Kuwamura and Galambos (1989) used the equation proposed by Akiyama (equation 4.3) 

and recommended the following values for VE

 T

T

I

G

E

E 2

1 V = for T ≤ TG 

 E G E T I

2

1 V = for (4.5) T ≥ TG

where is the integral of the square of the ground acceleration for the total duration of 

accelerogram 

E I

f t

∫ = f t

I E ug dt

0

2 && (4.6)

Using 40 accelerograms Fajfar et al. (1989) computed the earthquake input energy for the 

intermediate-periods (constant-velocity region) with 5% damping and strength ratios η = 

0.5-1.0, where η is the ratio of yield force Qy to mPGA , where PGA is the peak ground 

acceleration. They recommended to compute the input energy per unit mass as 

0.5 2 2.2(t ) (PGV)

m

E

di

i = (4.7)

in which tdi is the duration of strong motion defined by Trifunac and Brady (1975), and 

PGV is the peak ground velocity. They did not propose any formula for short and long 

periods, i.e. in the constant-acceleration and constant-displacement regions. 

As stated previously, Uang and Bertero (1990) proposed two procedures for computing 

the input energy per unit mass: one based on the absolute motion given by 

∫ = t

t g

ia u u d

m

E

0

&& & τ (4.8)

where u&&

t denotes the total acceleration (ut u ug && = &&+ && ), and the other based on relative 

motion given by 

∫ = − t

g

ir u u d

m

E

0

&& & τ (4.9)

12
Using five accelerograms (El Centro 1940, Taft 1952, Pacoima Dam 1971, Mexico City 

1985, and San Salvador 1986) they proposed the input energy for an SDOF structure with 

a ductility ratio μ = 5 and a 5% damping ratio be computed from

2 2 (1 0.12 ) ( ) 2

1

t PGV

m

E

di

ia = + (4.10)

 

Kuwamura et al. (1993) computed the equivalent velocityV in equation (4.3) as E

∫ = − f t

VE ug u dt

0

2 && & (4.11)

It should be noted that the energy expression under the square root sign is always 

positive. Using a different set of five accelerograms (El Centro 1940, Parkfield 1966, 

Hachinohe 1968, Pacoima Dam 1971, and SCT-Mexico 1985) they showed that for an 

undamped elastic structure, the equivalent velocity VE in equation (4.11) can be estimated 

by smoothing 1⁄2 , where FS is the Fourier spectra given by 2 (FS)

2

0

2

0 ( ) 

 

 +  

 = = ∫ ∫ f f t

g

t

VE FS ω u&&g Cosωt dt u&& Sinωt dt (4.12) 

in which ω is the circular frequency. Equation (4.11) used by Kuwamura and Galambos 

(1989), and equation (4.2) used by Zahrah and Hall (1984) are based on relative motion, 

see equation (4.9). 

Finally, Manfredi (2001) using 244 accelerograms examined the relation among the 

following three parameters: 

1) the equivalent number of yield excursions defined by Zahrah and Hall (1984) as Neq

( ) ( 1) ( 1) 2 − = − = − = μ y μ

hm

y y

hm

y m y

hm

eq k u

E

F u

E

F u u

E N (4.13) 

where u and are the maximum and yield deformations, respectively, and m uy um uy μ = /

is the ductility ratioa

, 

2) the response modification factor R , and 

3) the dimensionless seismic index given by d I

(PGA)(PGV)

I I E

d = (4.14)

 a According to Zahrah and Hall (1984) the minimum value of Neq is 1.0.

13
where IE is given by equation (4.6). For ductility ratios greater than 2, Manfredi 

proposed to estimate the input energy spectra in the constant-velocity region by 

2 0.23 ) ( ) 1

1 0.45 ( I PSV

m

E

d

i + − = μ

 (4.15)

where μ is the ductility ratio, and PSV denotes the pseudo-spectral velocity. Equations 

(4.6, 4.14, and 4.15) show that as the duration of strong motion increases, the input 

energy also increases. Equation (4.15) is similar to equation (4.1) where the term in the

parentheses reflects the influence of the duration of strong motion. Equation (4.15) also 

shows that as the ductility increases, the input energy decreases. 

4.2. The Influence of Ground Motion Characteristics on Energy 

Spectra 

The influences of the earthquake intensity, the duration of strong motion, and the 

frequency content on earthquake input energy are examined using 10 accelerogrmas with 

short duration (tdi shorter than 8 s) and 10 with long duration (tdi longer than 18 s) of 

strong motion, see Table 4.1. The duration tdi is computed using the definition proposed 

by Trifunac and Brady (1975), known as the intensity-based duration. They defined the 

duration as the time interval between the five and ninety-five percent contributions to the 

integral of the square of ground acceleration, see equation (4.6). 

Figure 4.1 shows the relative input energy spectra for a ductility ratio of 3 and a 5% 

damping ratio for the S00E component of El Centro with effective peak accelerations

EPA = 0.3g, 0.4g, and 0.6g, where EPA is the average of the acceleration spectra in the 

constant-acceleration region divided by 2.5 (ATC 3-06, 1978), see equation (1.1). 

According to the figure, as the effective peak acceleration increases, the input energy 

increases; indicating that the input energy is related to the intensity of ground motion. 

The energy ratios such as the maximum hysteretic energy to the maximum input energy 

Ehm/Eirm (see Figure 4.2) are not affected by EPA; indicating that the scaling of 

accelerograms does not influence the distribution of the earthquake input energy among 

the various energy components. 

Figure 4.3 presents the relative input and hysteretic energy spectra for the N65W

component of Temblor (tdi = 6 s) and the S69E component of Taft (tdi = 29 s) 

accelerograms scaled to an EPA = 0.4g for μ = 1, 3, and 5, and a 5% damping ratio. Both 

accelerograms have the same predominant period Tpe = 0.44 s, but the duration of strong 

motion for the Taft accelerogram is approximately 5 times longer than the duration for

the Temblor accelerogram. Figure 4.3.a shows that ductility has a minor effect on the 

input and hysteretic energies for both Taft and Temblor accelerograms. Figure 4.3 

indicates that on the average for μ = 1, 3, and 5, the energy spectra for the Taft 

accelerogram is approximately 8 times larger than the energy spectra for Temblor. The 

hysteretic energy demand for the Taft accelerogram is approximately 5 and 6 times larger 

14
15

Table 4.1. Strong motion records used in the analysis

1 2 3 4 5 6 7 8 9 10 11 

No Earthquake Date Station Comp. ML PGA (g) PGV (cm/s) tdi (s) 1 Tpa (s) 2 Tpe (s) 2 Dis. (km) 

Records with short strong motion duration

1 Northridge, CA

Jan 17, 1994

Pacoima Dam

Down Stream

175o 6.6 0.415 44.7 4 0.40 0.46 19.3 Highly jo

2 Northridge, CA

Jan 17, 1994

Pacoima Dam

Down Stream

265o 6.6 0.434 30.6 4 0.38 0.52 19.3 

3 Northridge, CA

Jan 17, 1994

Sylmar County 

Hospital Parking lot 

90o 6.6 0.604 76.9 7 0.52 2.28 15.8 

4 Northridge, CA

Jan 17, 1994

Sylmar County 

Hospital Parking lot 

360o 6.4 0.843 128.9 5 0.36 1.56 15.8 

5 Parkfield, CA 

June 27, 1966

Temblor California N65W 6.0 0.269 14.5 6 0.26 0.44 43.9 Alluviu

6 Parkfield, CA 

June 27, 1966

Temblor California S25W 6.0 0.347 22.5 5 0.38 0.38 43.9 

7 San Fernando, CA 

Feb 9, 1971 

Pacoima Dam S16E 6.7 1.170 113.2 7 0.38 1.50 8.5 Highly jo

8 San Fernando, CA 

Feb 9, 1971 

Pacoima Dam S74W 6.7 1.075 57.6 7 0.42 0.42 8.5 

9 San Francisco,CA 

Mar 22, 1957

Golden Gate Park S80E 5.3 0.104 4.6 3 0.22 0.24 11.5 

10 San Francisco,CA 

Mar 22, 1957

Golden Gate Park N10E 5.3 0.084 4.91 4 0.26 0.26 11.5 
16

Table 4.1. Continued

1 2 3 4 5 6 7 8 9 10 11 

No Earthquake Date Station Comp. ML PGA (g) PGV (cm/s) tdi (s) 1 Tpa (s) 2 Tpe (s) 2 Dis. (km) 

Records with long strong motion duration

11 Valparaiso, Chile 

Mar 3, 1985

Llolleo - basement

1-story building 

N10E 7.83 0.711 41.5 36 0.22 0.52 N.A. Sandsto

12 Valparaiso, Chile 

Mar 3, 1985

Llolleo - basement

1-story building 

100o 7.83 0.445 23.3 41 0.14 0.42 N.A. 

13 Imperial Valley, CA 

May 18, 1940

El Centro S00E 6.7 0.348 33.4 24 0.26 0.86 11.5 Alluvium 

th

14 Imperial Valley, CA 

May 18, 1940

El Centro S90W 6.7 0.214 36.9 25 0.48 2.14 11.5 

15 Kern County, CA 

July 21, 1952

Taft, Lincoln School

Tunnel

S69E 7.2 0.179 17.7 29 0.44 0.44 41.4 40ft all

ceme

16 Landers, CA 

June 28,1992 

Joshua Tree 

Fire Station 

0o 7.3 0.273 28.1 31 0.42 1.18 13.7 

17 Landers, CA 

June 28,1992 

Joshua Tree 

Fire Station 

90o 7.3 0.283 42.7 28 0.72 0.74 13.7 

18 Michoacan, Mexico 

Sep 19, 1985

Mexico City 

SCT 

S00E 8.5 0.100 38.7 71 2.04 2.06 350 Thick d

19 Tabas, Iran 

Sep 16, 1978

Tabas 74o 7.4 0.879 97.8 18 0.24 0.88 1.2 

20 Tabas, Iran 

Sep 16, 1978

Tabas 344o 7.4 0.937 114.1 18 0.20 4.72 1.2 

1 tdi is the duration of strong motion defined by Trifunac and Brady (1975). 2 Tpa and Tpe are the predominant periods based on the elastic acceleration and input energy spectra, see Figure 4.4. 3 Surface magnitude Ms. 
0

4

0 5 Period (s)

In

p

ut ener

g

y Eir (

m2/s2)

EPA=0.6g

EPA=0.4g

EPA=0.3g

El Centro S00E, t di = 24 s

Figure 4.1. Relative input energy for ductility ratio μ = 3 and a damping ratio ξ = 5% for the S00E 

component of El Centro, the Imperial Valley, CA earthquake of May 18, 1940 scaled to EPA = 0.3g, 

0.4g, and 0.6g. Parameters for Bouc-Wen Model are A = 1, β = γ = 0.5, n = 20, α = 0.02

17
0

1

0 5 Period (s)

Ehm/Eirm

EPA=0.3g

EPA=0.4g

EPA=0.6g

El Centro S00E, t di = 24 s

Figure 4.2. Ratio of the maximum hysteretic energy to the maximum input energy for ductility ratio 

μ = 3 and a damping ratio ξ = 5% for the S00E component of El Centro, the Imperial Valley, CA 

earthquake of May 18, 1940 scaled to EPA = 0.3g, 0.4g, and 0.6g. Parameters for Bouc-Wen Model 

are A = 1, β = γ = 0.5, n = 20, α = 0.02

18
0

2.5

0 5 Period (s)

Eir(m2/s2

)

Taft S69E μ=1

Taft S69E μ=3

Taft S69E μ=5

Temblor N65W μ=1

Temblor N65W μ=3

Temblor N65W μ=5

(a) E ir

t di = 29 s

t di = 6 s

0

2.5

0 5 Period (s)

Eh (m2/s2

)

Taft S69E μ=3

Taft S69E μ=5

Temblor N65W μ=3

Temblor N65W μ=5

(b) E h

t di = 29 s

t di = 6 s

Figure 4.3. (a) Earthquake input energy Eir and (b) hysteretic energy Eh for the N65W component of 

Temblor, Park-field, CA earthquake of June 27, 1966 (tdi = 6 s) and the S69E component of Taft

Lincoln School Tunnel, Kern County, CA earthquake of July 21, 1952 (tdi = 29 s) for μ = 1, 3, and 5, 

and a damping ratio ξ = 5% scaled to an EPA = 0.4g. Both accelerograms have the same

predominant period Tpe = 0.44 s. Parameters for Bouc-Wen Model are: A = 1, β = γ = 0.5, n = 20, and

α = 0.02. The hysteretic energy for μ = 1 is not shown in Figure 4.3.b since it is null.

19
than the hysteretic energy demand for the Temblor accelerogram for μ = 3 and 5, 

respectively. Figure 4.3.b indicates that as the duration of strong motion increases, the

hysteretic energy demand also increases. 

For the 20 accelerograms used, the predominant periods Tpe computed from the input

energy spectra are larger than Tpa computed from the acceleration spectra, see columns 9 

and 10 of Table 4.1. Predominant periods Tpa and Tpe primarily depend on the soil 

conditions. The average Tpe/Tpa ratio for the twenty accelerograms is 3.3. Figure 4.4

shows the normalized acceleration and input energy spectra for three accelerograms with 

short duration of strong motion (Figures 4.4.a – 4.4.c), and three accelerograms with long 

duration (Figures 4.4.d – 4.4.f). Since Tpa is generally different from Tpe, the strength and 

energy demands do not necessarily occur at the same period. Therefore, both strength 

and energy demands should be examined in design. 

Figure 4.5 shows the elastic input energy spectra for the 90o

 component of Sylmar and the 

S16E component of Pacoima Dam records scaled to an EPA = 0.4g. Both accelerograms

have the same duration of strong motion (tdi = 7 s). The bracketed duration of the scaled 

accelerograms are also close (tdb = 12 s for Sylmar accelerogram and tdb = 11 s for 

Pacoima Dam accelerogram). Figure 4.5 indicates that frequency content has a 

significant influence on input energy spectra, even for the same intensity and duration of 

strong motion. Figures 4.2 – 4.5 conclude that for a given effective peak acceleration the 

influence of the duration of strong motion on the input energy is as significant as the 

influence of the frequency content. 

4.3. The Influence of Structural Properties on Energy Spectra 

McKevitt et al. (1980), Zahrah and Hall (1984), Akiyama (1985), and Nakashima et al.

(1996) believe that ductility and damping do not have a significant influence on the 

earthquake input energy. It should be noted that these studies used 4, 8, 1, and 3 

accelerograms, respectively. Housner (1956), and Berg and Thomaides (1966) believe 

that in designing a structure to satisfy the energy demand, the elastic input energy 

computed from 1⁄2m(PSV)

2

 can be used conservatively in lieu of inelastic input energy. 

Kato and Akiyama (1982), and Leelataviwat et al. (1999) used Housner’s 

recommendation, equation (4.1), to develop an energy-based design method for steel 

buildings. This section examines the influences of structural properties on the energy 

spectra. 

Figures 4.6.a - 4.6.c and 4.6.d – 4.6.f show the relative input energy Eir, equation (4.9), 

for three accelerograms with short and three accelerograms with long duration of strong 

motion for ductility ratios μ = 1, 2, 3, and 5, and a damping ratio ξ = 5%. It is observed 

that ductility has a significant influence on the magnitude and frequency content of the 

input energy spectra for the 1994 Sylmar, 1971 Pacoima Dam, and 1985 Mexico City 

accelerograms (Figures 4.6.a, 4.6.c, and 4.6.f), particularly in the vicinity of the 

predominant periods Tpe (definition of Tpe is illustrated in Figure 4.4). Although the 

elastic input energy Eir does not provide a conservative estimate of the inelastic input

20
0

1

0 5 Period (s)

Sa/Samax

Eir/Eirmax

T pa T pe

(a) Sylmar 360o, t di =5 s

0

1

0 5 Period (s)

Sa/Samax

Eir/Eirmax

T pa T pe

 (c) Pacoima Dam S16E

 t di = 7 s

0

1

0 5 Period (s)

Sa/Samax

Eir/Eirmax

T pe

T pa

(b) Temblor California

 N65W, t di =6 s

0

1

0 5 Period (s)

Spectral ratio

Sa/Samax

Eir/Eirmax

T pe

(d) El Centro S00E, t di =24 s

0

1

0 5 Period (s)

Spectral ratio

Sa/Samax

Eir/Eirmax

T pa & T pe

(f) Mexico City SCT

S00E t di = 71 s

0

1

0 5 Period (s)

Spectral ratio

Sa/Samax

Eir/Eirmax

T pa

T pe

(e) Llolleo N10E

 t di =36 s

T pa

tio al ra ectr Sp io rat tral Spec ratio ral ect Sp

Figure 4.4. Predominant periods Tpa and Tpe from the elastic acceleration and input energy spectra, 

respectively, for three accelerograms with short duration of strong motion (a - c), and three

accelerograms with long duration (d - f) in Table 4.1. Parameters for Bouc-Wen Model are: A = 1, β

= γ = 0.5, n = 20, and α = 0.02 

21
0

2.5

0 5 Period (s)

Eir(m2/s2

)

Sylmar 90 deg

Pacoima Dam S16E

 μ = 1

T pe = 1.5 s

T pe = 2.3 s

Figure 4.5. Elastic input energy spectra for the 90o component of Sylmar County Hospital Parking

Lot, Northridge, CA earthquake of January 17, 1994 and the S16E component of Pacoima Dam, San

Fernando, CA earthquake of March 22, 1971 scaled to an EPA = 0.4g. Parameters for Bouc-Wen 

Model are: A = 1, β = γ = 0.5, n = 20, and α = 0.02

22
0

6

0 5 Period (s)

(a) Sylmar 360o, t di =5 s

0

2

0 5 Period (s)

Input e

nergy (m2/s2)

μ=1

μ=2

μ=3

μ=5

(d) El Centro S00E, t di =24 s

0

0.3

0 5 Period (s)

(b) Temblor California N65W, t di =6 s

0

8

0 5 Period (s)

In

p

ut energ

y (

m2/s2)

(e) Llolleo N10E, t di =36 s

0

6

0 5 Period (s)

In

p

ut

(c) Pacoima Dam S16E

 t di = 7 s

0

14

0 5 Period (s)

In

p

ut energ

y (

m2/s2)

(f) Mexico City S00E

 t di = 71 s

s2) 2/ y (m erg t e

n

npu I ) 2/s2 (

m erg

y en

p

ut In s2) (

m2/

y energ

Figure 4.6. Relative input energy Eir for ductility ratios μ = 1, 2, 3, and 5, and a damping ratio ξ = 

5% for three accelerograms with short (a - c), and three accelerograms with long duration of strong 

motion (d - f) in Table 4.1. Parameters for Bouc-Wen Model are: A = 1, β = γ = 0.5, n = 20, and α = 

0.02 

23
energy for Pacoima Dam and Mexico City accelerograms (Figures 4.6.c and 4.6.f), it 

generally overestimates the inelastic input energy. 

Figure 4.7 shows that the relative input energy for a ductility ratio μ = 3, a post- to preyield stiffness ratio α = 0.02, and damping ratios ξ = 0, 2, 5, 10, 20, and 40% for three 

accelerograms with short duration of strong motion (a - c), and three with long duration

(d - f) in Table 4.1. The reason for including damping ratios larger than 5% is that large 

damping ratios may are encountered when using supplemental damping devices in a 

structure. Figure 4.7 shows that for damping ratios smaller than 5%, damping has little 

effect on the input energy. For damping ratios greater than 5%, on the other hand,

damping has a significant influence on the input energy spectra, particularly for periods 

close to Tpe. A damping ratio ξ = 40% may reduce the input energy by approximately 

50% for periods close to Tpe. For very long periods as the damping increases, the input 

energy increases. 

Figure 4.8 shows the energy ratios for a ductility ratio μ = 3 and damping ratios 0, 2, 5,

10, 20, and 40% subjected to the 360o

 component of the 1994 Sylmar record with tdi = 5 

s. It is observed that as the damping increases, the Ehm/Eirm and (Eh/Eir)m ratios decrease, 

Figures 4.8.c and 4.8.d. Figure 4.8.c indicates that for a structure with a 5% damping ratio 

approximately 75% of the input energy is dissipated through hysteretic action, and for a 

structure with a 40% damping only 20% of the input energy is dissipated through 

hysteretic action. The figure indicates that damping significantly influences the 

distribution of input energy among the energy components. 

Fajfar et al. (1989), and Uang and Bertero (1990) recommended that the input energy be 

used as a damage index. Results of the study by Bruneau and Wang (1996) and this 

study (see Figure 4.7) indicate that damping ratios smaller than 5% have a minor 

influence on the input energy. On the other hand, damping affects the damage potential 

of structures to a large degree (see Figures 4.8.c and 4.8.d). Therefore, the input energy 

should not be used as a sole damage index. Earthquake input energy, however, can be

used as a measure of the intensity of ground motion. Peak ground acceleration PGA, 

effective peak acceleration EPA defined by Equation (1.1), and Spectrum Intensity SI

defined by equation (1.2) have all been used in measuring the intensity of ground motion, 

but they do not account for the influence of the duration. Earthquake input energy, on the 

other hand, accounts for the duration of strong motion, frequency content, and structural 

properties and is more meaningful than PGA, EPA, and SI for measuring the intensity.

The maximum ordinate of the elastic input energy spectrum with 5% damping may be 

used for scaling the records to the same intensity. 

The hysteretic behavior of a structure depends on the material and the structural system 

such as moment resisting frame, braced frame, etc. Two cases of material non-linearities 

are considered in this study: (a) bilinear behavior with A = 1, β = γ = 0.5, and n = 20, and 

α = 0.02 proposed by Wen (1976) for a moment resisting steel frame (MRF), and (b) 

hysteretic behavior with A = 1.4704, β = 0.042, γ = -0.121, n = 2, and α = 0.2325 

proposed by Loh and Chung (1993) for an eccentrically braced steel frame (EBF). Figure 

4.9 shows the hysteresis loops for an MRF and an EBF structure with Tn = 2.0 s, μ = 3, 

24
0

3

0 5 Period (s)

(a) Sylmar 360o, t di =5 s

0

3

0 5 Period (s)

Input energy (

m2/s2)

ξ=0% ξ=2%

ξ=5% ξ=10%

ξ=20% ξ=40%

(d) El Centro S00E, t di =24 s

0

0.14

0 5 Period (s)

(b) ParkfieldN65W, t di =6 s

0

7

0 5 Period (s)

In

p

ut energ

y (

m2/s2)

(e) Llolleo S00E, t di =36 s

0

4

0 5 Period (s)

(c) Pacoima Dam S16E, t di = 7 s

0

7

0 5 Period (s)

In

p

ut energ

y (

m2/s2)

(f) Mexico City S00E, t di = 71 s

s2) (

m2/ erg

y en t n

p

u I s2) m2/ y ( erg

ut en

n

p I s2) m2/ ( erg

y en

ut n

p I

Figure 4.7. Relative input energy Eir for a ductility ratio of 3, and damping ratios ξ = 0, 2, 5, 10, 20,

and 40% for three accelerograms with short duration of strong motion (a - c), and three 

accelerograms with long duration (d - f) in Table 4.1. Parameters for Bouc-Wen Model are: A = 1, β

= γ = 0.5, n = 20, and α = 0.02

25
0

1

0 1 2 345

Period (s)

Edism/Eirm

ξ = 0%

ξ = 2%

ξ = 5%

ξ = 10%

ξ = 20%

ξ = 40%

(f) E dism /E irm

0

1

0 1 2 3 4 5

Period (s)

Ehm/Eirm

ξ = 0%

ξ = 2%

ξ = 5%

ξ = 10%

ξ = 20%

ξ = 40%

(c) E hm /E irm

0

1

0 1 2 345

Period (s)

(Eh/Eir)m

(d) (E h /E ir) m

0

1

0 1 2 3 4 5

Period (s)

Ekrm/Eirm

ξ = 0%

ξ = 2%

ξ = 5%

ξ = 10%

ξ = 20%

ξ = 40%

0

1

0 1 2 345

Period (s)

Esm/Eirm

ξ = 0%

ξ = 2%

ξ = 5%

ξ = 10%

ξ = 20%

ξ = 40%

(b) E sm /E irm

0

1

0 1 2 3 4 5

Period (s)

Edm/Eirm

(e) E dm /E irm

(a) E krm /E irm

Figure 4.8. Energy ratio spectra for bilinear behavior with a ductility ratio μ = 3 and damping ratios 

0%, 2%, 5%, 10%, 20%, and 40% subjected to the 360o

 component of Sylmar County Hospital

Parking Lot Station, Northridge earthquake of January 17, 1994. (a) Ekrm/Eirm, (b) Esm/Eirm, (c)

Ehm/Eirm, (d) (Eh/Eir)m, (e) Edm/Eirm, and (f) Edism/Eirm. Parameters for Bouc-Wen Model are: A = 1, β = 

γ = 0.5, n = 20, and α = 0.02

26
-1

0

1

-50 0 50

Displacement (cm)

fs/W

(a) MRF

T n = 2.0 s

μ = 3

I

-1

0

1

-50 0 50

Displacement (cm)

fs/W

(b) EBF

T n = 2.0 s

μ = 3

II

Figure 4.9. Force-displacement diagram for an SDOF MRF and an SDOF EBF with Tn = 2.0 s, a

ductility ratio of 3, and a 5% damping ratio subjected to the 360o

 component of Sylmar County 

Hospital Parking Lot, Northridge, CA earthquake of January 17, 1994. Parameters for Bouc-Wen

Model are: (a) A = 1, β = γ = 0.5, n = 20, and α = 0.02, and (b) A = 1.4704, β = 0.042, γ = -0.121, n =2,

and α = 0.2325

27
and a 5% damping ratio subjected to the 360o

 component of the Sylmar record. The total 

input energy transmitted by that record to the MRF and EBF structures at a period of 2 s 

are approximately 1.9 and 4.5 m2

/s2 (see Figure 4.10) 

Figure 4.10 shows the input energy spectra for a ductility ratio of 5, a 5% damping ratio,

and the two types of hysteretic behavior for the 360o

 component of the Sylmar record and 

the S00E component of the Mexico City record. The figure indicates that the shape of 

the hysteresis loop has a significant influence on the magnitude and variation of the input 

energy spectra. 

The results of this study indicate that the influences of structural properties on earthquake 

input energy are as important as the influences of ground motion characteristics. 

Nonetheless, the empirical equations proposed by Akiyama (1985), Kuwamura and 

Galambos (1989), Fajfar et al. (1989), Uang and Bertero (1990), and Kuwamura et al.

(1993) for estimating the earthquake input energy are based on the ground motion 

characteristics only. The equation proposed by Housner (1956) does not account for the 

influence of the duration of strong motion. The procedure proposed by Manfredi (2001) 

for the constant-velocity region of response spectra accounts for the influences of the 

ground motion characteristics as well as structural properties in estimating the earthquake 

input energy. 

A procedure for computing the earthquake input energy from the response spectra by 

accounting for the soil type is needed. A large number of accelerograms recorded on 

different soil should be used to compute the energy spectra for various hysteretic 

behaviors.

28
0

10

012 345

Period (s)

In

p

ut energ

y

Eir (m2/s2)

MRF

EBF

(a) Sylmar 360o 

t di = 5 s

E ir = 1.9 m2

/s2

E ir = 4.5 m2

/s2

0

10

012 3 4 5

Period (s)

In

p

ut energ

y

Eir (m2/s2)

MRF

EBF

(b) Mexico City, SCT S00E

t di = 71 s

Figure 4.10. Relative input energy Eir for a ductility ratio of 3, damping ratio ξ = 5%, and two 

hysteretic behaviors for (a) the 360o

 component of Sylmar County Hospital Parking Lot, Northridge,

CA earthquake of January 17, 1994, and (b) the S00E component of Mexico City earthquake of

September 19, 1985. Parameters for Bouc-Wen Model are (I) for MRF structure: A = 1, β = γ = 0.5,

n = 20, and α = 0.02, and (II) for EBF structure A = 1.4704, β = 0.042, γ = -0.121, n =2, and α = 0.2325

29

5. HYSTERETIC ENERGY DEMAND 

In a seismic design procedure based on energy, one should limit the structural damage by 

providing sufficient ductility and energy dissipation capacity by hysteretic action and/or 

damping in the structure. The damage potential is associated with the maximum 

hysteretic energy demand during the excitation and during the largest yield excursion (see 

Figures 3.1 – 3.3). The hysteretic energy demand can be computed from the input energy 

spectra if the ratio of the maximum hysteretic energy to the maximum input energy 

Ehm/Eirm is known. To examine the relationship between hysteretic energy and damage 

potential three energy ratios are considered: 1) the maximum ratio of hysteretic to input 

energy (Eh/Eir)m generally occurring during the largest yield excursion, 2) the ratio of the 

maximum hysteretic energy to the maximum input energy Ehm/Eirm occurring at the end 

of the excitation, and 3) the equivalent number of yield excursions Neq, see equation 

(4.13). 

Figure 5.1 shows (Eh/Eir)m, Ehm/Eirm, and Neq for a ductility ratio μ = 3 and a 5% damping 

ratio for the records with short and long duration of strong motion. Figures 5.1.a to 5.1.d 

indicate that, on the average, (Eh/Eir)m are approximately 20% larger than Ehm/Eirm

regardless of the period of structure. Ground motion characteristics have a minor

influence on the (Eh/Eir)m ratio (Figures 5.1.a and 5.1.b) and a major influence on the 

(Eh/Eir)m ratio (Figures 5.1.c and 5.1.d). For example, for an SDOF structure with Tn = 

1.0 s (Figure 5.2) subjected to the 360o

 component of 1994 Sylmar record (tdi = 5 s), 

approximately 90% of the total hysteretic energy is dissipated during the first (largest)

yield excursion where (Eh/Eir)m = 0.77 and Ehm/Eirm = 0.67. Similarly for a structure with 

Tn = 2.0 s (Figure 5.3), 70% of Ehm is dissipated in the first (largest) yield excursion 

where (Eh/Eir)m = 0.71 and Ehm/Eirm = 0.69. Therefore, when a flexible structure (long 

period) is subjected to a short duration earthquake, the damage potential during the 

largest yield excursion, indicated by (Eh/Eir)m, may be more important than the damage

potential due to the total number of yield excursions indicated by Ehm/Eirm. 

Figures 5.1.e and 5.1.f show that generally for the same ductility and damping, the

equivalent number of yield excursions Neq is larger for the records with long duration of

strong motion than for the records with short duration, indicating a larger damage 

potential for the long duration records. For the two longest duration records, the 1985 

Mexico City - S00E (tdi = 71 s) and 1985 Llolleo - 100o

 (tdi = 41 s), Neq is the largest in 

the vicinity of the predominant period where Tpe = 2.0 s and 0.52 s for the Mexico City 

and Llolleo records, respectively (see Figure 5.1.f). Figure 5.1 indicates that the 

structures with Tn = 1.0 to 2.5 s and Tn = 0.2 to 0.7 s have the potential for significant

damage when subjected to the Mexico City and Llolleo records, respectively. One 

cannot make a definitive conclusion from Figures 5.1.e and 5.1.f, however, for the 

damage potential of structures subjected to the other 18 accelerograms. 

31
0 1 2 3 4 5

Period (s)

Ehm/Eirm

(c) Ehm /Eirm

SD

0

1

0 1 2 3 4 5

Period (s)

Ehm/Eirm

(d) Ehm /Eirm

LD

0 1 2 3 4 5

Period (s)

(Eh/Eir)m

(a) (Eh /Eir ) m

SD

0

1

0 1 2 3 4 5

Period (s)

(Eh/Eir)m

(b) (Eh /Eir ) m

LD

0

20

0 1 2 3 4 5

Period (s)

Neq

(e) N eq

SD

0

20

40

0 1 2 3 4 5

Period (s)

Neq

(f) N eq

LD

Mexico City

SCT-S00E Llolleo-100o

1

0

1

0

40

Figure 5.1. (a, b) Maximum ratio of the hysteretic to input energy (Eh/Eir)m, (c, d) Ratio of the 

maximum hysteretic energy to the maximum input energy Ehm/Eirm, (e, f) Equivalent number of yield 

excursions Neq for records with short and long durations for a ductility ratio μ = 3 and a 5% damping

ratio. Parameters for Bouc-Wen Model are: A = 1, β = γ = 0.5, n = 20, and α = 0.02. 

32
0

1

0 10 20 30 40

Time (s)

Eh/Ehm and 

Eh/Eir

Eh/Eir

Eh/Ehm

90% of E hm in the

first yield excursion

E hm = 0.93 (m

2 /s 2 )

E irm = 1.39 (m

2 /s 2 )

E hm /E irm = 0.67

(E h /E ir ) m = 0.77

50

Figure 5.2. Time-histories of Eh/Ehm and Eh/Eir for a bilinear SDOF structure with Tn = 1.0 s, a

ductility ratio μ = 3, and 5% damping subjected to the 90o

 component of Sylmar County Hospital 

Parking Lot, Northridge, CA earthquake of January 17, 1994. Parameters for Bouc-Wen Model are

A = 1, β = γ = 0.5, n = 20, α = 0.02

33
Figure 5.3 demonstrates that, in general, the energy ratios considered provide different 

information about the damage potential of a structure. The ratio (Eh/Eir)m reflects the 

damage due to the largest yield excursion and is independent of the duration of strong 

motion and period of structure. The energy ratios Ehm/Eirm and Neq, on the other hand, 

reflect the damage potential due to cumulative inelastic deformation for the entire 

duration of accelerogram. The equivalent number of yield excursions Neq depends on the

duration of strong motion and the period of structure, whereas the Ehm/Eirm ratio is 

independent of both only for periods less than 1 s. For quantifying the damage potential, 

the Ehm/Eirm and (Eh/Eir)m ratios may be used to define a damage index to address the 

damage potential for the largest yield excursion and the total number of yield excursions

during the earthquake. 

Figures 5.4.a and 5.4.b show the mean (Eh/Eir)m and Ehm/Eirm ratios, respectively, for the 

20 accelerograms for 5% damping and ductility ratios μ = 2, 3, and 5. In general, as the 

ductility increases both ratios increase, indicating a larger dissipation of input energy 

through hysteretic action. Figure 5.4.b can be used to estimate the hysteretic energy 

spectra from input energy spectra if the relationship among the Ehm/Eirm ratio, the duration

of strong motion, and the period of structure is determined. A large number of 

accelerograms recorded on different soil should be used and energy spectra for various 

hysteretic behaviors should be computed to develop a procedure for estimating the 

hysteretic energy demand. 

34
0

1

0 10 20 30 40 5

Time (s)

Eh/Ehm and 

Eh/Eir

Eh/Eir

Eh/Ehm

70% of E hm in the

first yield excursion

E hm = 1.28 (m

2 /s 2 )

E irm = 1.86 (m

2 /s 2 )

E hm /E irm = 0.69

(E h /E ir ) m = 0.71

0

Figure 5.3. Time-histories of Eh/Ehm and Eh/Eir for a bilinear SDOF structure with Tn = 2.0 s, a

ductility ratio μ = 3, and 5% damping subjected to the 90o

 component of Sylmar County Hospital 

Parking Lot, Northridge, CA earthquake of January 17, 1994. Parameters for Bouc-Wen Model are

A = 1, β = γ = 0.5, n = 20, α = 0.02 

35
0

1

0 5 Period (s)

ax th tio h

re

inp

ut e

n

e

rgy (Eh/Eir)

m

μ=2

μ=3

μ=5

(a) mean (E h /E ir) m

0

1

0 5 Period (s)

R

atio of th

e m

a

xim

u

m hysteretic e

n

ergy

to th

e m

a

xim

u

m inp

ut e

n

ergy Ehm/Eir

m

μ=2

μ=3

μ=5

(b) mean E hm /E irm

tic to

yste of

e ra m of imu

M

Figure 5.4. The mean (Eh/Eir)m and Ehm/Eirm spectra for bilinear behavior with 5% damping and 

ductility ratios μ = 2, 3, and 5 for 20 accelerograms. Parameters for Bouc-Wen Model are: A = 1, β = 

γ = 0.5, n = 20, and α = 0.02. 

36
6. ENERGY RATIOS IN AN MDOF STRUCTURE 

Studies by McKevitt et al. (1980) and Nakashima et al. (1996) suggest that the energy 

transmitted to an MDOF structure can be estimated from the energy transmitted to an

equivalent SDOF structure. Once the input and the hysteretic energies for a structure are 

estimated, one needs to distribute the energies through the height of the structure. 

Sufficient strength and energy dissipation capacity should be provided in structural 

members to limit the damage potential of the structure to acceptable thresholds (refer to 

Vision 2000). This Chapter examines the distribution of earthquake input energy and its 

components in a five-story building using the 20 accelerograms in Table 4.1 scaled to an 

effective peak acceleration EPA = 0.4g. 

The structural properties for the building are as follows: all floors have a weight W = 

1000 KN. The pre-yield stiffness of the first story is K1 = 228,898 KN/m and the 

stiffnesses for stories 2 – 5 are reduced linearly by 10%, i.e. Kj/K1 = 0.9, 0.8, 0.7, and 0.6 

for j = 2 to 5, respectively. Damping is assumed proportional to stiffness. Considering a

5% damping in the first mode of vibration of the elastic structure, the damping coefficient 

for the first story is C1 = 3643 KN-s/m and the damping distribution through the height is 

Cj/C1 = 0.9, 0.8, 0.7, and 0.6 for j = 2 to 5, respectively. The yield deformation for each 

story varies proportionally to the stiffness (Yang et al. 1992), where uyj/uy1 = 0.9, 0.8, 0.7, 

and 0.6 for j = 2 to 5, respectively, and uy1 is the yield deformation for the first story. The 

pre-yield periods of the building are Tn = 0.50, 0.17, 0.11, 0.08, and 0.07 s. The BoucWen parameters (Appendix A) used in the analyses are A = 1, β = γ = 0.5, n = 6, α = 0.02 

for all floors. The energy balance equation for MDOF structures is presented in 

Appendix B. 

Figure 6.1 shows the total hysteretic and input energies for the five-story building with 

uy1 = 0.5 cm vs. the intensity-based duration computed for the 20 records. As expected, 

the figure shows that the total input and hysteretic energies for the records with long 

duration are greater than those for the records with short duration. The standard

deviations of Eh for the records with short and long duration are 0.1 MN.m and 0.4 

MN.m, respectively, and the standard deviations of Eir for the short and long duration 

records are 0.1 MN.m and 0.5 MN.m, respectively. The large dispersion from the means 

in Figure 6.1 is caused by the influence of the frequency content on input energy, i.e. the 

input and hysteretic energies for a structure with a period approximately equal to the 

predominant period Tpe, see Table 4.1, are larger than those for a structure with period 

different than the predominant period. Figure 6.1 indicates that the duration of strong 

motion has a significant influence on energy demand. 

Table 6.1 shows the inter-story ductility, normalized story shear, energy distribution, and 

the ratios (Eh/Eir)m, Ehm/Eirm, and Edm/Eirm for each story when the building is subjected to

the S00E component of El Centro accelerogram scaled to an EPA = 0.4g. The yield

deformation for the first story is uy1 = 0.5 cm. Table 6.1 shows that the ductility demand 

37
0

1

2

0 25 50

t di(s)

Eh (MN.m)

E h,ave=0.11

SD

E h,ave=0.48

LD

(b) E h

0

1

2

0 25 50

t di(s)

Eir (MN.m)

E ir,ave=0.76

LD

E ir,ave=0.18 (a) E ir

SD

75

75

Figure 6.1. Total hysteretic and input energies for the five-story building with uy1 = 0.5 cm subjected 

to the 20 accelerograms scaled to an EPA = 0.4g. Parameters for Bouc-Wen Model are A = 1, β = γ = 

0.5, n = 6, α = 0.02 

38
Table 6.1. Inter-story ductility, normalized story shear, percentage of energy in each story, and the

energy ratios (Eh/Eir)m, Ehm/Eirm, and Edm/Edm for the five-story building subjected to the S00E 

component of El Centro, the Imperial Valley, CA earthquake of May 18, 1940 scaled to an EPA = 

0.4g. 

1 2 3 4 5 6 7 8 9 10 11 12

Story μj Qj/Qb

Ekr 

(%)

Ed 

(%)

Es 

(%)

Eh 

(%)

Edis 

(%)

Eir 

(%) (Eh/Eir)m Ehm/Eirm Edm/Eirm

5 1.3 0.32 31 36 5 6 1 2 0.34 0.09 0.51

4 3.3 0.49 32 30 14 12 12 13 0.74 0.57 0.43

3 4.6 0.68 23 23 25 21 31 29 0.76 0.67 0.33

2 4.2 0.85 12 16 29 29 33 31 0.76 0.65 0.35

1 3.2 1.00 3 16 28 36 24 25 0.76 0.58 0.42

39
is the largest in the third story (column 2), whereas the hysteretic energy demand is the 

largest in the first story (column 7). The ductility ratio does not consider the influence of 

the duration, whereas the hysteretic energy does. Therefore, it is reasonable that in

design both ductility and energy demands be examined for each story. It is observed that 

for higher stories the relative kinetic energy Ekr is larger (column 4) since it is computed 

from the relative velocity, whereas elastic strain energy Es is smaller (column 6) since it 

is computed from the driftb

 which for a shear building is smaller in higher stories. The 

maxima of Eh/Eir are presented in column 10 of Table 6.1 which are approximately equal 

for the four lower stories. The fifth story behaves approximately elasticly (μ = 1.3 in

column 2). Only 9% of the input energy is dissipated through hysteresis (column 11) and 

51% of the energy is dissipated through damping. It is also observed that (Eh/Eir)m is 

generally larger than Ehm/Eirm for each story and the difference between the two ratios 

increases for higher stories which have smaller ductility demands. 

Table 6.2 shows the influence of the yield deformation on the maximum inter-story 

ductility, base shear, and total energy parameters for the five-story building subjected to

the S00E component of the 1940 El Centro accelerogram scaled to an EPA = 0.4g. It is

observed that as the yield deformation decreases ductility demand increases (column 2), 

and base shear decreases significantly (column 3), whereas the total dissipated and input 

energies remain approximately the same (columns 8 and 9). Nonetheless, ductility has a 

pronounced influence on the distribution of input energy among energy components, 

specially for ductility demands less than 3 (columns 4 - 7). 

Figure 6.2 shows the relative displacement, ductility demand, normalized story shear, the 

energy ratio Ehm/Eirm, and the distribution of the hysteretic and input energies though the 

height of the 5-story building with yield deformations uy1 = 0.3, 0.5, 0.7, 1.0, and 1.5 cm, 

subjected to the S00E component of the 1940 El Centro accelerogram scaled to an EPA = 

0.4g. Also presented are the results for the elastic case. The figure indicates that as the 

yield deformation decreases the relative displacement for each story decreases, and the 

ductility and hysteretic energy demands increase. It is observed that structural nonlinearity has a small effect on the distribution of story shear, hysteretic energy, and input 

energy for this building, see Figures 6.2.c, 6.2.e, and 6.2.f. Even though displacement is 

largest at the top, the ductility and hysteretic energy demands are largest in the lower 

three stories.

Figure 6.3 compares the percentage of energy distributed through the height of the fivestory building, with a yield deformation for the first story uy1 = 0.5 cm, subjected to the 

20 accelerograms scaled to an EPA = 0.4g. The figure indicates that in general, the 

distribution of energy through the height of the building is not affected by ground motion 

characteristics. On the average, 23, 28, 27, 15, and 7% of the total input energy is 

transmitted to stories 1 to 5, respectively. Similarly, 22, 30, 31, 16, and 1% of the total 

hysteretic energy is dissipated in stories 1 to 5, respectively. While the above 

percentages provide rough estimates and trend in distribution and dissipation of energy, a 

large number of structures and records should be examined to arrive at a rational formula 

for distribution of energy through the height of a structure. 

 b

 The difference between two consecutive floor displacements. 

40
Table 6.2. Influence of yield deformation on maximum inter-story ductility μmax, base shear Qb, and 

total energy for the five-story building subjected to the S00E component of El Centro, the Imperial

Valley, CA earthquake of May 18, 1940 scaled to an EPA = 0.4g. 

1 2 3 4 5 6 7 8 9

uy1 

(cm) 

μmax Qb 

(KN) 

Ekrt 

(KN-m)

Edt 

(KN-m)

Est 

(KN-m)

Eht 

(KN-m)

Edist 

(KN-m)

Eirt 

(KN-m)

Elastic 1.0 5038 193 583 162 0 583 584

1.5 1.8 3439 82 443 70 202 645 645

1.0 2.4 2333 52 342 31 300 642 642

0.7 3.8 1642 53 272 16 339 611 611

0.6 4.2 1420 54 245 12 343 588 588

0.5 4.6 1196 54 215 9 347 562 562

0.4 5.0 968 52 185 6 355 540 540

0.3 8.3 762 49 160 3 364 524 524

0.2 12.8 550 43 144 2 353 497 497

41
0

1

2

3

4

5

0 1 E hm/E irm

Story

(d) E hm /E irm

Elastic

0

1

2

3

4

5

0 100

Hysteretic energy E h (%)

Story

uy=0.3

uy=0.5

uy=0.7

uy=1.0

uy=1.5

Elastic

(e) Distribution of E h

Elastic

0

1

2

3

4

5

0 100

Input energy E ir (%)

Story

uy=0.3

uy=0.5

uy=0.7

uy=1.0

uy=1.5

Elastic

(f) Distribution of E ir

1

2

3

4

0 2 4 6 8 10

Relative displacement (cm)

Story

uy =.3

uy=0.5

uy=0.7

uy=1.0

uy=1.5

Elastic

(a) u rel

0

1

2

3

4

0 2 4 6 8 10

Ductility demandμ

Story

(b) μ 

Elastic

0

1

2

3

0 1

Normalized story shear

Story

uy=0.3

uy=0.5

uy=0.7

uy=1.0

uy=1.5

Elastic

(c) Q j /Q b

5

0

5

5

4

Figure 6.2. (a) Relative displacement urel, (b) ductility demand μ, (c) normalized story shear, (d)

Ehm/Eirm, (e) distribution of Eh though the height, and (f) distribution of Eir through the height for the

5-story building with yield deformations of the first story uy1 = 0.3, 0.5, 0.7, 1.0, and 1.5 cm subjected 

to S00E component of El Centro, the Imperial Valley, CA earthquake of May 18, 1940 scaled to an 

EPA = 0.4g. Also presented are the results for elastic behavior. Parameters for Bouc-Wen Model 

are: A = 1, β = γ = 0.5, n = 20, and α = 0.02

42
0 25 50 75 100

Distribution of relative kinetic energy

Story

(a) E kr

0

1

2

3

4

5

0 25 50 75 100

Distribution of elastic strain energy

Story

(b) E s

0 25 50 75 100

Distribution of hysteretic energy

Story

(c) E h

0

1

2

3

4

5

0 25 50 75 100

Distribution of damping energy

Story

(d) E d

0

1

2

3

4

5

0 25 50 75 100

Distribution of input energy

Story

(f) E ir

0 25 50 75 100

Distribution of dissipated energy

Story

(e) E dis

5

4

3

2

1

0

5

4

3

2

1

0

5

4

3

2

1

0

Figure 6.3 Distribution of energy through the height of a 5-story building with uy1 = 0.5 cm subjected 

to 20 accelerograms scaled to an EPA = 0.4g. Parameters for Bouc-Wen Model are: A = 1, β = γ = 

0.5, n = 20, and α = 0.02 

43

7. DISTRIBUTION OF INPUT ENERGY IN STRUCTURES WITH 

BASE-ISOLATION, SUPPLEMENTAL DAMPING AND SEMIACTIVE CONTROL 

Base isolation, Figure 7.1.b, supplemental damping, and semi-active control are used to 

reduce the damaging effects of earthquakes. Base isolation reduces the motion 

transmitted to the structure and the base shear by increasing the period of structure.

Supplemental damping reduces the hysteretic energy demand on the structure by 

dissipating a portion of the input energy. Semi-active control combines the features of 

active and passive control. Similar to passive systems, control forces are generated by 

the response of the building, and similar to active systems, controllers are used to monitor 

feedbacks and apply the control forces. Unlike active systems which require a large 

power source to generate the control forces, semi-active systems require a smaller power 

source. 

 

Semi-active control (Figure 7.1.c) includes active variable stiffness and active variable 

damping. In variable stiffness control, the stiffness of the building is adjusted to prevent 

resonance of the building from the excitation. In variable damping control, supplemental 

energy dissipation devices such as friction and fluid dampers are used to reduce the 

structural response. This chapter presents the energy balance equation for structures with 

base-isolation and semi-active control, Section 7.1, and the energy distribution for four 

five-story buildings with: fixed-base, base-isolation, supplemental damping, and semiactive control in Section 7.2. 

7.1. Energy Equations for Buildings with Base-Isolation and SemiActive Control

The energy balance equation for a base-isolated building with a base mass mb (Figure 

7.1.b) is similar to a fixed-base building (Figure 7.1.a) with an additional degree of

freedom for the base isolation, see equation B.2 in Appendix B. Assuming that the 

building undergoes a rigid body motion, the period of the base-isolated building can be 

approximated as 

b

b

j

j

b k

m m

T

+

=

∑=

5

1

( )

2π and 

b

b T

π

ω

2 = (7.1) 

and the damping ratio of the base isolation system as 

45
0 30

m5 m5

5 5 5 

m4 m4

4 4 4 

m3 m3

3 3 3 

m2 m2

2 2 2 

m1 m1

1 1 1 

mb

b

(a) (b) (c)

60

Figure 7.1. (a) Five-story building with fixed base, (b) with base-isolation, and (c) with supplemental 

damping or semi-active control subjected to the S00E component of El Centro, the Imperial Valley, 

CA earthquake of May 18, 1940 scaled to an EPA = 0.4g

46
b b

j

j

b

b

m m

c

ω

ξ

2( )

5

1

+

=

∑=

 (7.2)

The energy balance equation for a building with supplemental damping (Figure 7.1.c) is 

similar to a fixed-base building, see equation B.2 in Appendix B. The total damping of 

the structure (inherent and supplemental) should be used in the analysis. 

The governing differential equation of motion for a non-linear n-story shear building with 

m semi-active dampers (Figure 7.1.c) is (see Yang et al. 1992) 

F (t) F (t) F (t) u (t) DF (t) i + d + s = − g + c ξ && (7.3)

where is an - dimensional inertia force vector, is an - dimensional damping 

force vector, is an - dimensional non-linear restoring force vector, 

Fi n

s

Fd n

F n ξ is an - 

dimensional mass vector, is ground acceleration,

n

ug && D is an n × m matrix defining the 

locations of control forces, and denotes an -dimensional control force vector. The 

inertia force vector can be written as 

Fc m

F M u&& i = where M is an n × n mass matrix and 

is an - dimensional acceleration vector. The damping force can be written as

 where is an damping matrix and u is an - dimensional velocity 

vector. The restoring force, however, is a non-linear function of u and u . 

u&&

F

n

C u& d = C n × n & n

&

Using the state-space representation, equation (7.3) can be written as 

z(t) g[z(t)] B u(t) H u (t) g & = + + && (7.4)

in which the 2n - dimensional vector g[z(t)] is given by 













− + = − [ ]

( ) [ ( )] 1 M Fd Fsu t

g z t & (7.5)

and z(t) is 











 = ( )

( ) ( )

u t

u t

z t & (7.6)

The control input matrix B is 

 

47




 



 = − M D

B 1

0

 (7.7)

and the disturbance input vector H is 











 = − ξ

1

0

M

H (7.8)

For the linear quadratic regulator algorithm (LQR), the optimal control force in 

equation (7.3) is selected by minimizing the performance index 

F (t) c

J

J [ ] z t Q z t F t R F t dt f t

c

T

c

T

∫ = +

0 ( ) ( ) ( ) ( ) (7.9) 

where is the duration of accelerogram, f t Q = qI is a 2n × 2n symmetric positive 

semi-definite weighting matrix ( q is a scalar and I is unit matrix), and R = rI is an

m × m positive definite weighting matrix (r is a scalar and I is unit matrix). By 

linearizing the hysteretic behavior of the structure at the initial equilibrium state and 

minimizing the performance index subject to the constraint of equation (7.4), the control 

forces are obtained as 

( ) 0.5 ( ) ( ) 1 F t R B P z t G z t T

c = − = − (7.10)

where G is an m × 2n feedback gain matrix and P is a 2n × 2n matrix obtained from 

the solution of the classical Riccati equation 

2Q z + Λ(z)P z + P z + P z& = 0 & (7.11)

where 

Λ(z) = ∂g(z)/ ∂z (7.12)

Transposing equation (7.3), multiplying both sides by du = u& dt and integrating, one 

obtains the energy balance equation as 

F t du F t du F t du u t du F t D du u T T

c

u

g

u T

s

u T

d

T u

∫ i ∫ ∫ ∫ ∫ + + = − +

0 0 0 0 0 ( ) ( ) ( ) ξ && ( ) ( ) (7.13) 

The energy corresponding to the controller is 

48
E F D u dt T t T

c c & ∫ = 0 (7.14)

Other energy vectors are defined in Appendix B. Equation (7.13) can be written as 

Ekr + Ed + Es + Eh = Eir + Ec (7.15)

where each term is an n -dimensional vector. 

7.2. Numerical results 

The lead-core rubber isolation bearing used in the example has a period Tb = 2.0 s and a 

damping ratio ξb = 10%. The weight of the base is equal to the weight of individual 

floors Wb =1000 KN, the stiffness of the base isolation system kb = 6036 KN/m, and the

damping coefficient = 384 KN-s/m. The Bouc-Wen parameters for the lead-core 

rubber bearing are A = 1, β = γ = 0.5, n = 3, and α = 0.6 (Yang et al., 1992). For the 

building with supplemental damping one damper is used at each story and the total 

damping (inherent and supplemental) is 40%. For the building with semi-active control 

one damper is used at each story. The weight parameters used for 20 accelerograms are r 

= 10

b c

-5 and q = 10+5. As q increases, the control force increases. The yield deformation 

for the first story is uy1 = 0.5 cm for all buildings. 

Figure 7.2, 7.3, and 7.4 show the percentage of energy distributed through the height of 

the five-story building with base isolation, supplemental damping, and semi-active 

control subjected to the 20 records scaled to an EPA = 0.4g. Figures 6.3, 7.2, 7.3, and 7.4 

conclude that ground motion characteristics have a minor influence on the distribution of 

energy through the height of the buildings. Figure 7.2 shows that base isolation is 

effective for structures subjected to all accelerograms, except the Mexico City 

accelerogram since the period of base-isolation (Tb = 2.0 s) is close to the predominant 

period of the accelerogram (Tpe = 2.0 s). Therefore, using base-isolation for a 5 story 

building in Mexico City is detrimental, while it is effective in reducing the ductility and

energy dissipation demand for building in other locations. 

The average percentage of hysteretic energy, input energy, and story shear in each story 

for the four buildings subjected to the 20 accelerograms scaled to an EPA = 0.4g are 

given in Table 7.1 and plotted in Figure 7.5. The figure indicates that the energy 

distribution for the buildings with fixed base, supplemental damping, and semi-active 

control are similar, whereas the energy distribution for the building with base isolation is 

significantly affected by the isolator, i.e. the majority of the input energy is absorbed and 

dissipated in the isolator. The story shear distribution is also similar for the four 

buildings. 

Table 7.2.a presents the maximum drift, the maximum relative displacement, the ductility

ratio, and the maximum story shear, and Table 7.2.b shows the maximum energy 

49
0 25 50 75 100

Distribution of relative kinetic energy

Story

(a) E kr

0

1

2

3

4

5

0 25 50 75 100

Distribution of elastic strain energy

Story

(b) E s

0 25 50 75 100

Distribution of hysteretic energy

Story

(c) E h

0

1

2

3

4

5

0 25 50 75 100

Distribution of damping energy

Story

(d) E d

0

1

2

3

4

5

0 25 50 75 100

Distribution of input energy

Story

(f) E ir

0 25 50 75 100

Distribution of dissipated energy

Story

(e) E dis

5

4

3

2

1

0

5

4

3

2

1

0

5

4

3

2

1

0

Figure 7.2 Distribution of energy through the height of a 5-story building with base isolation (10%

damping, Tb = 2.0 s, and uyb = 4.0 cm) with uy1 = 0.5 cm subjected to 20 records scaled to an EPA = 

0.4g. Parameters for Bouc-Wen Model for the stories are: A = 1, β = γ = 0.5, n = 20, and α = 0.02 and

for base isolation system are: A = 1, β = γ = 0.5, n = 3, and α = 0.6 

50
0 25 50 75 100

Distribution of relative kinetic energy

Story

(a) E kr

0

1

2

3

4

5

0 25 50 75 100

Distribution of elastic strain energy

Story

(b) E s

0 25 50 75 100

Distribution of hysteretic energy

Story

(c) E h

0

1

2

3

4

5

0 25 50 75 100

Distribution of damping energy

Story

(d) E d

0

1

2

3

4

5

0 25 50 75 100

Distribution of input energy

Story

(f) E ir

0 25 50 75 100

Distribution of dissipated energy

Story

(e) E dis

5

4

3

2

1

0

5

4

3

2

1

0

5

4

3

2

1

0

Figure 7.3. Distribution of energy through the height of a 5-story building with supplemental 

damping (total damping 40%) with uy1 = 0.5 cm subjected to 20 records scaled to an EPA = 0.4g. 

Parameters for Bouc-Wen Model for the stories are: A = 1, β = γ = 0.5, n = 20, and α = 0.02 

51
0 25 50 75 100

Distribution of relative kinetic energy

Story

(a) E kr

0

1

2

3

4

5

0 25 50 75 100

Distribution of elastic strain energy

Story

(b) E s

0 25 50 75 100

Distribution of hysteretic energy

Story

(c) E h

0

1

2

3

4

5

0 25 50 75 100

Distribution of damping energy

Story

(d) E d

0

1

2

3

4

5

0 25 50 75 100

Distribution of input energy

Story

(f) E ir

0 25 50 75 100

Distribution of dissipated energy

Story

(e) E dis

5

4

3

2

1

0

5

4

3

2

1

0

5

4

3

2

1

0

Figure 7.4. Distribution of energy through the height of a 5-story building with semi-active control (r 

= 10-5 and q = 10+5) with uy1 = 0.5 cm subjected to 20 records scaled to an EPA = 0.4g. Parameters for

Bouc-Wen Model for the stories are: A = 1, β = γ = 0.5, n = 20, and α = 0.02 

52
Table 7.1. The average distribution percentage of hysteretic energy, input energy, and story shear, in 

percent, in each story of the five-story building with fixed-base, base isolation, supplemental

damping, and semi-active control subjected to 20 accelerograms scaled to an EPA = 0.4g. 

Table 7.1.a Distribution of hysteretic energy Eh (%)

1 2 3 4 5

Story Fixed 

Base 

Base 

Isolation 

Suppl.

Damp. 

Semi 

Active 

5 1 0 1 1

4 16 0 8 12

3 31 0 22 30

2 30 1 33 34

1 22 1 36 23

Base - 98 - -

Table 7.1.b Distribution of input energy Eir (%)

1 2 3 4 5

Story Fixed 

Base 

Base 

Isolation 

Suppl.

Damp. 

Semi 

Active 

5 7 6 6 7

4 15 6 11 20

3 27 6 20 21

2 28 6 28 19

1 23 6 35 23

Base - 71 - -

Table 7.1.c Distribution of story shear Q (%)

1 2 3* 4 5

Story Fixed 

Base 

Base 

Isolation 

Suppl.

Damp. 

Semi 

Active 

5 32 26 29 31

4 50 49 48 49

3 67 68 64 66

2 84 85 82 83

1 100 100 100 100

*

 The distribution of story shear for the building with base isolation is given for the super structure. 

53
0

1

2

3

4

5

0 Eh 100

St

ory

Fixed Base

Base Isolation

Suppl. Damping

Semi Active

(a) Distribution of E h

0

1

2

3

4

5

0 E ir 100

Stor

y

Fixed Base

Base Isolation

Suppl. Damping

Semi Active

(b) Distribution of E ir

0

1

2

3

4

5

0 1 Q

Story

Fixed Base

Base Isolation

Suppl. Damping

Semi Active

(c) Distribution of Q

00

Figure 7.5. Distribution of (a) hysteretic energy Eh, (b) input energy Eir, and (c) story shear Q

distributed in the stories of the 5-story building with fixed base, base isolation, supplemental 

damping, and semi-active control subjected to 20 records scaled to an EPA = 0.4g. Parameters for 

Bouc-Wen Model for the stories are: A = 1, β = γ = 0.5, n = 20, and α = 0.02. Note that for the 

building with base isolation the percentage is given for the base shear of the super structure.

54
Table 7.2. (a) Maximum drift, maximum relative displacement, ductility ratio, and maximum story shear for each story, and 

maximum hysteretic, maximum controller, and maximum input energies for the four buildings (with fixed-base, base-isolationand semi-active control) subjected to the S00E component of El Centro, the Imperial Valley, CA earthquake of May 18, 1940 sc

Table 7.2.a Response parameters 

with fixed-base with base-isolation with supplemental damping with se

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1

Story Drift urel μ Q Drift urel μ Q Drift urel μ Q Drift u

(cm) (cm) (KN) (cm) (cm) (KN) (cm) (cm) (KN) (cm) (c

5 0.39 6.72 1.3 409 0.11 20.97 1.0 157 0.31 3.29 1.0 387 0.22 24 1.15 6.40 3.3 625 0.19 20.87 1.0 307 0.56 2.98 1.6 577 0.46 23 1.85 5.26 4.6 865 0.24 20.69 1.0 446 0.71 2.43 1.8 761 0.73 22 1.91 3.48 4.2 1077 0.28 20.45 1.0 579 0.82 1.73 1.8 965 0.84 11 1.62 1.62 3.2 1273 0.32 20.18 1.0 729 0.91 0.91 1.8 1192 0.75 0

Base - - - - 19.87 19.87 5.0 816 - - - - - 

55 Table 7.2.b Energy parameters 

with fixed-base with base-isolation with supplemental damping with se

1 2 3 4 5 6 7 8 9 10 11 12 

Story Ed Eh Eir Ed Eh Eir Ed Eh Ec Eir Ed E

(KN-m) (KN-m) (KN-m) (KN-m) (KN-m) (KN-m) (KN-m) (KN-m) (KN-m) (KN-m) (KN-m) (K

5 10 2 20 0 0 21 12 0 13 1 4 31 41 72 0 0 21 39 3 42 5 3 54 108 163 0 0 21 69 8 77 8 2 63 116 179 0 0 20 98 12 110 11 1 61 83 144 1 0 20 122 14 136 11 

Base - - - 254 104 358 - - - - 

Total 218 351 578 255 104 360 340 37 378 36 2

a Total input energy transmitted to stories 1 to 5 are 103 KN-m.
parameters: damping, hysteretic, controller, and input for each story of the four buildings

subjected to the S00E component of El Centro, the Imperial Valley, CA earthquake of 

May 18, 1940 scaled to an EPA = 0.4g. To have a larger force in controllers, the weight 

parameters used in the analysis are selected as r = 10-5 and q = 10+6. Except for the baseisolated building, the total input energy is equal to the sum of the maximum input 

energies of all stories. For the base-isolated building, all stories behave elastically, and 

the base isolation system behaves inelastically. The maximum input energy for the

stories occurs in the initial stages of the excitation, whereas the maximum input energy 

for the base isolation system occurs at the end of the excitation. Consequently, one 

cannot add up the input energies measured at different times to obtain the total input

energy. The total input energy for stories 1 to 5 of the base-isolated building is 103 KNm (see Table 7.2.b). The total input energy transmitted to the buildings with baseisolation, supplemental damping, and semi-active control are approximately 20%, 65%, 

and 25% of the total input energy transmitted to the building with fixed base,

respectively. 

For the base-isolated building (Tb = 2.0 s) subjected to the El Centro accelerogram (Tpa = 

0.26 s and Tpe = 0.86 s, see Table 4.1), the base shear (see columns 5 and 9 of Table 7.2.a 

and Figure 7.6.c), and the total input energy (columns 4 and 7 of Table 7.2.b) are 

approximately 60% of those for the fixed base building. The story drifts (columns 2 and 

6 of Table 7.2.a) of the base-isolated building are less than 30% of the story drift for the 

fixed-base building. The base-isolated building behaves like a rigid body (Figure 7.6.a) 

and all stories remain elastic during the excitation (column 8 of Table 7.2.a; Figures 

7.6.b, 7.6.d, and 7.6.e). The results indicate that the input energy is dissipated in the base 

isolation system (columns 5 and 7 of Table 7.2.b). 

The base shear for the building with semi-active control is approximately 90 percent of

the base shear for the fixed-base building, whereas the input energy is approximately 20 

percent. The maximum control forces generated in the buildings with semi-active control 

is 288 KN. 

Table 7.3 shows the distribution of energy among various energy components for the four 

buildings subjected to the S00E component of El Centro, the Imperial Valley, CA 

earthquake of May 18, 1940 scaled to an EPA = 0.4g. The results indicate that providing

base isolation, supplemental damping, or semi active dampers has a significant influence 

on the energy demand. 

Finally, the first story drift and total input energy for the four buildings subjected to the

S00E component of the 1940 El Centro accelerogram scaled to an EPA = 0.4g are 

compared in Figures 7.7 and 7.8, respectively. It is observed that both the drift and input 

energy of the buildings with base-isolation, supplemental damping, and semi-active 

control are less than those for the fixed-base building. According to Figure 7.7, baseisolation is the most effective system in reducing the base shear and drift for this building 

subjected to El Centro accelerogram. Results conclude that base–isolation, supplemental 

damping, and semi-active control are all effective in reducing the hysteretic energy 

demand for this building subjected to El Centro accelerogram. 

56
0

1

2

3

4

5

0 1

Ehm /Eirm

Story

(d) Ehm /Eirm

0

1

2

3

4

5

0 1

(Eh /Eir) m

Story

(e) (Eh /Eir) m

0

1

2

3

4

5

0 1

Edm /Eirm

Story

(f) Edm /Eirm

0

1

2

3

4

5

0 500 1000 1500

Story shear (KN )

Story

(c) Q j

0

1

2

3

4

5

0 2 4 6

Ductility demand

Story

(b) μ 

0

1

2

3

4

5

0 5 10 15 20 25

Relative displacement (cm )

Story

Fixed base

Base Isolation

Suppl. Damping

Semi-Active Control

(a) u rel

Figure 7.6. (a) Relative displacement, (b) ductility demand, (c) story shear, (d) Ehm/Eirm, (e) (Eh/Eir)m, 

and (f) Edm/Eirm for the five-story buildings: fixed-base, base-isolated, semi-active with large control 

force, and semi-active with small control force with uy1 = 0.5 cm and ξ1 = 5% subjected to the S00E

component of El Centro, the Imperial Valley earthquake of May 18, 1940 scaled to an EPA = 0.4g. 

Parameters for Bouc-Wen Model are: A = 1, β = γ = 0.5, n = 20, and α = 0.02, and for the base 

isolation system are: A = 1, β = γ = 0.5, n = 3, and α = 0.6

57
Table 7.3. The energy ratios Ekr/Eir, Ed/Eir, Es/Eir, Eh/Eir, and Ec/Eir for the the four buildings (a)

fixed-base, (b) base-isolated, (c) semi-active with small control forces, (d) and semi-active with large 

control forces subjected to the S00E component of El Centro, the Imperial Valley, CA earthquake of

May 18, 1940 scaled to an EPA = 0.4g. 

Table 7.3.a The building with fixed-base 

1 2 3 4 5 6

Story Ekrm/Eirm Esm/Eirm Edm/Eirm Ehm/Eirm Edism/Eirm

1 0.90 0.51 0.03 0.09 0.60

2 0.26 0.43 0.02 0.57 1.00

3 0.09 0.33 0.01 0.67 1.00

4 0.04 0.35 0.02 0.65 1.00

5 0.01 0.42 0.02 0.58 1.00

Total 0.10 0.38 0.02 0.62 1.00

Table 7.3.b The building with base isolation system 

1 2 3 4 5 6

Story Ekrm/Eirm Esm/Eirm Edm/Eirm Ehm/Eirm Edism/Eirm

1 1.00 0.00 0.00 0.00 0.00

2 0.99 0.01 0.01 0.00 0.01

3 0.98 0.02 0.03 0.00 0.02

4 0.97 0.02 0.04 0.00 0.02

5 0.96 0.03 0.06 0.00 0.03

Base 0.05 0.71 0.15 0.29 1.00

Total 0.33 0.71 0.16 0.29 1.00

Table 7.3.c The building with supplemental damping 

1 2 3 4 5 6

Story Ekrm/Eirm Esm/Eirm Edm/Eirm Ehm/Eirm Edism/Eirm

1 0.32 0.93 0.04 0.02 0.95

2 0.08 0.93 0.02 0.07 1.00

3 0.03 0.90 0.02 0.10 1.00

4 0.01 0.89 0.02 0.11 1.00

5 0.00 0.90 0.02 0.10 1.00

Total 0.03 0.90 0.02 0.10 1.00

Table 7.3.d The building with semi-active control 

1 2 4 3 5 6 7

Story Ekrm/Eirm Esm/Eirm Edm/Eirm Ehm/Eirm Edism/Eirm Ecm/Eirm

1 0.37 0.12 0.03 0.00 0.95 0.82

2 0.16 0.21 0.05 0.07 1.00 0.72

3 0.06 0.25 0.05 0.19 1.00 0.56

4 0.02 0.28 0.06 0.22 1.00 0.50

5 0.01 0.32 0.08 0.14 1.00 0.54

Total 0.08 0.26 0.06 0.15 1.00 0.59

58
 

0

0 10 20 30 40 5

Time (s)

Total input energy Eir (K

N-m)

Fixed-base, Eirm=578 KN-m

Base-islation, Eim=360 KN-m

Suppl. Damping, Eirm=378 KN-m

Semi-Active Control, Eirm=140 KN-m

-1.8

0

1.8

0 10 20 30 40 5

Time (s)

First story drift umax (cm)

Fixed-base, umax=1.22 cm

Base-Isolation, umax=0.27 cm

Suppl. Damping, umax=0.89 cm

Semi-Active Control, umax=0.74 cm

0

Figure 7.7. Drift time-histories for the first story of the buildings with fixed-base, base-isolation,

supplemental damping, and semi-active control subjected to the S00E component of El Centro, the

Imperial Valley, CA earthquake of May 18, 1940 scaled to an EPA = 0.4g. Parameters used for

Bouc-Wen Model for the stories are A = 1, β = γ = 0.5, n = 20, α = 0.02, and for the base isolation

system are A = 1, β = γ = 0.5, n = 3, α = 0.6. The total damping of the building with supplemental

damping is 40%. The weighting matrices for semi-active building are defined by r =10-5 and q = 10+6. 

600

0

Figure 7.8. Total input energy time-histories for the five-story buildings with fixed-base, baseisolation, supplemental damping, and semi-active control subjected to the S00E component of El 

Centro, the Imperial Valley, CA earthquake of May 18, 1940 scaled to an EPA = 0.4g. Parameters 

used for Bouc-Wen Model for the stories are A = 1, β = γ = 0.5, n = 20, α = 0.02, and for the base 

isolation system are A = 1, β = γ = 0.5, n = 3, α = 0.6. The total damping of the building with 

supplemental damping is 40%. The weighting matrices for semi-active building are defined by r =10-

5

 and q = 10+6. 

59

8. CONCLUSIONS 

Using 20 accelerograms, the distribution of earthquake input energy in a one and a fivestory building with fixed base, based isolation system, and semi-active control system are 

examined. The following observations and conclusions are presented: 

8.1. SDOF structures: 

1. Earlier studies on a few accelerograms indicated that structural properties: 

ductility, damping, and the post- to pre-yield stiffness ratio for bilinear behavior

do not have a significant influence on the earthquake input energy. Using 20 

accelerograms this study, however, shows that ductility, damping, and the shape 

of the hysteresis loop have a major influence on the input energy spectra and its 

distribution among various energy components. Results indicate that in 

estimating the energy parameters, the influences of the structural properties are as

significant as the influences of ground motion characteristics. 

2. This study shows that as the effective peak acceleration increases, the input 

energy also increases; indicating that the input energy is related to the intensity of

ground motion. The energy ratios such as the ratio of the maximum hysteretic

energy to the maximum input energy Ehm/Eirm are not affected by the effective

peak acceleration. Therefore, scaling an accelerogram does not change the 

distribution of earthquake input energy among energy components. Results 

indicate that the influence of the duration of strong motion on the input and 

hysteretic energies is as significant as the influence of frequency content, 

particularly for non-linear structures. As the ductility ratio increases, the 

influence of the duration of strong motion on the input energy spectra becomes

more significant, particularly in the vicinity of the predominant period Tpe. 

3. For the 20 accelerograms used, the predominant period Tpa (computed from the

acceleration spectra) is different from Tpe (computed from the energy spectra) 

indicating that the strength and energy demands do not necessarily occur at the 

same period. Therefore, in developing an energy-based design approach one 

should examine both strength and energy demands. 

4. The study concludes that the elastic input energy spectra Eir do not necessarily 

provide a conservative estimate of the inelastic input energy spectra. 

5. Three energy ratios based on hysteretic energy were compared: the maximum 

ratio of the hysteretic to input energy (Eh/Eir)m, the ratio of the maximum 

hysteretic energy to the maximum input energy Ehm/Eirm, and the equivalent 

number of yield excursions Neq. It is found that the (Eh/Eir)m ratio reflects the

damage potential associated with the largest yield excursion, and the Ehm/Eirm and 

61
Neq ratios reflect the damage potential associated with the total number of yield

excursions and the cumulative inelastic deformation for the entire duration of 

accelerogram. The study shows that (Eh/Eir)m is independent of the duration of 

strong motion and period of structure, whereas Ehm/Eirm is independent of both 

only for periods less than 1 s. Results show that as the duration of strong motion 

increases the equivalent number of yield excursions Neq increases; indicating 

more seismic vulnerability. Nonetheless, one cannot make a definitive 

conclusion from Neq spectra for comparing the damage potential of various 

structures. 

6. It is observed that the number and the rate of yield excursions depend on the 

ground motion characteristics (intensity, frequency content, and duration of strong 

motion), and structural properties (period of structure, ductility, damping, and

hysteretic behavior). The damage potential for the largest yield excursion is as

significant as the total yield excursion in evaluating the damage potential of

structures. 

7. The results indicate that small damping ratios (less than 5%) have a minor 

influence on the input energy and a major influence on the damage potential of 

structures. Therefore, input energy should not be considered as a sole damage

index. For damping ratios greater than 5%, however, damping reduces the input

energy in the vicinity of the predominant period Tpe. 

8. Earthquake input energy reflects the intensity of earthquake ground motion and is 

more appropriate for measuring the intensity than using the effective peak

acceleration EPA (equation 1.1), Spectrum Intensity SI (equation 1.2), and 

Accelerogram Intensity IE (equation 4.6) since Eir considers both the 

characteristics of ground motion and the structural properties. The maximum

ordinate of the elastic input energy spectrum with 5% damping may be used for 

scaling the records to the same intensity. 

8.2. Five-story building with fixed base, base-isolation, supplemental damping, 

and semi-active control: 

1. For a given effective peak acceleration, the distribution of energy and story shear 

through the height of the five-story building (with fixed base, base-isolation, 

supplemental damping, and semi-active control) is mostly independent of the 

duration of strong motion and the frequency content of the accelerograms. 

2. Total drift and total input energy for the building with base-isolation, 

supplemental damping, and semi-active control are significantly less than those 

for the fixed-base building. 

62
3. The ductility and hysteretic energy demands do not necessarily occur in the same

story. The story shear is the largest in the first story, whereas ductility and energy 

dissipation may not be the largest in the first story. 

4. Base-isolation is effective in reducing the damage potential of the 5-story 

structure by dissipating most of the earthquake input energy in the base isolation 

system. Nonetheless, if the period of the base-isolation system is close to the

predominant period of the ground motion, the hysteretic energy demand and 

consequently the damage potential for the building with base-isolation will be

larger than those for the building with fixed base. The predominant period of the 

ground motion primarily depends on the soil conditions. It also depends on the 

fault and path characteristics and epicentral distance (specially in near fault 

regions). 

5. For the 5-story building with fixed base subjected to the S00E component of the

El Centro accelerogram, as the yield deformation decreases, base shear decreases

significantly, and the total input energy remains approximately the same. It is 

found that ductility has a pronounced influence on the distribution of input energy

among energy components. 

63

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68
APPENDIX A. BOUC-WEN MODEL 

Different models are used to characterize the nonlinear behavior of structures such as 

Ramberg-Osgood (Ramberg and Osgood, 1943), Bouc-Wen (Bouc, 1967; Wen, 1976 and 

1980), Penzien and Liu (Penzien and Liu, 1969), Takeda (Takeda, 1970), Sina (Saiidi and 

Sozen, 1979), and Q-Histeresis (Saiidi and Sozen, 1979), Otani (Otani, 1981), 

Vinodgradov-Pivovarov (Vinodgradov and Pivovarov, 1986). The Bouc-Wen model 

which is extensively used by researchers (see Wen, 1989) to represent the forcedeformation relationship is used in this study. 

For the Bouc-Wen model, Figure A.1, the equation of motion can be written as 

y ug u&&+ u& + u + − u z = −&& 2

0

2

0 0 0 2ξ ω αω (1 α)ω (A.1)

where 

u = displacement, 

uy = yield displacement, 

ug && = ground acceleration, 

ξ 0= viscous damping ratio, 

ω 0 = pre-yield natural frequency, 

α = post- to pre-yield stiffness ratio. 

In the above equation, z is a non-dimensional hysteretic parameter which is related to 

displacement u through the following first-order non-linear differential equation 

( ) 1 n 1 n

y z& = u Au& − β u& z z& −γu& z − −

 (A.2)

Parameters γ and β define the shape of the hysteretic loop (softening or hardening); 

parameter A controls the restoring force amplitude and tangent stiffness, and n defines the 

smoothness of the transition from elastic to inelastic range in the force-deformation 

relationship. This model can be used for a variety of materials. For example A = 1, γ = 

β = 0.5, n = 20 and α = 0 can approximately represent the elasto-plastic behavior. 

69
m

(1-α) k uy z

α k u

u ug &&

c u& m

(1-α) k uy z

α k u

u ug &&

c u&

Figure A.1 Single degree of freedom system used for Bouc-Wen model 

70
APPENDIX B. EARTHQUAKE INPUT ENERGY FOR MDOF

 STRUCTURES 

The equation of motion for an degree of freedom shear building (Figure 7.1.a) can be

written as 

n

F (t) F (t) F (t) Mr u (t) i d s g + + = − && (B.1) 

where is an -dimensional inertia force vector, is an -dimensional damping

force vector, is an -dimensional non-linear stiffness force vector, 

Fi n Fd n

Fs n M is the mass 

matrix, r is an n -dimensional influence vector, and u denotes ground acceleration. g &&

Transposing the matrix equation (B.1), multiplying both sides by and

integrating, we obtain the energy balance equation as 

du = u& dt

F du F t du F t du Mr u t du u

g

u T

s

u T

d

u T

∫ i ∫ ∫ ∫ + + = − 0 0 0 0 ( ) ( ) && ( ) (B.2) 

where u is an n -dimensional relative displacement vector. The relative kinetic energy is 

E Mu t du u M u

T u

kr &&( ) 1⁄2 & & 0 = = ∫ (B.3)

The damping energy is given by 

E F t u dt t T

d d & ∫ = 0 ( ) (B.4)

The absorbed energy vector is given by 

h s

t T Ea = Fs t u dt = E + E ∫ & 0 ( ) (B.5)

the elastic strain energy for the jth story of building is 

j

sj

sj k

Q E

2

2

= (B.6)

71
where and are the shear force and elastic stiffness of the jth story, respectively. 

Once the elastic strain energy is obtained, the hysteretic energy can be computed from

equation (B.5). Finally, the relative input energy is obtained as 

Qsj j k

E Mr u t u dt t

ir g && & ∫ = − 0 ( ) (B.7)

Therefore, the vector equation (B.2) can be written as 

Ekr + Ed + Es + Eh = Eir (B.8)

72
APPENDIX C. LIST OF SYMBOLS 

aμ A regression parameter depending on the ductility and the duration of 

strong motion

A Parameter of Bouc-Wen model describing tangent stiffness 

bμ A regression parameter depending on the ductility and the duration of 

strong motion

B Control input matrix 

c Viscous damping coefficient of SDOF structure 

cb Viscous damping coefficient for base-isolation system

cμ A regression parameter depending on the ductility and the duration of 

strong motion

C Viscous damping coefficient of each story of the structure

dμ A regression parameter depending on the ductility and the duration of 

strong motion

Ea Absorbed energy 

Ec Energy corresponding to energy 

Ed Equivalent viscous damping energy 

Edm Maximum equivalent viscous damping energy 

Edis Energy dissipated through damping and hysteretic behavior 

Edism Maximum dissipated energy through damping and hysteretic behavior 

Edist Total dissipated energy through damping and hysteretic behavior 

Edt Total equivalent viscous damping energy 

Eh Hysteretic (yielding or plastic strain) energy 

Ehm Maximum hysteretic energy

Eht Total hysteretic energy 

Eia Absolute input energy 

Eir Relative input energy 

Eirm Maximum relative input energy 

Eirt Total relative input energy 

Ekr Relative kinetic energy 

Ekrm Maximum relative kinetic energy 

Ekrt Total relative kinetic energy 

Es Elastic strain energy 

Esm Maximum elastic strain energy 

Est Total elastic strain energy 

EPA Effective peak ground acceleration

fd Damping force 

fi Inertia force

fs Restoring force 

Fc Control force 

Fd Damping force 

Fi Inertia force

Fs Restoring force

73
Fy Yield force 

FS Fourier Spectra 

g Gravity acceleration of ground

G Feedback gain matrix

H Disturbance input vector

I Unit matrix 

IE Intensity of accelerogram 

Id Seismic Index 

J Performance index 

k Stiffness of the SDOF structure 

kb Stiffness of the base isolation system 

K Stiffness for each story of the structure

m Mass of the SDOF structure 

mb Mass of the base for a base-isolated building 

n A parameter for Bouc-Wen model controlling the smoothness of transition 

from elastic to inelastic region of the force-displacement curve 

Neq Equivalent number of yield excursions

P A matrix for the solution of Riccati equation 

PGA Peak ground acceleration

PGV Peak ground velocity

q A scalar 

Q Story shear; a weight matrix in Chapter 7 

Qb Base shear 

r A scalar 

R Response modification factor; a weight matrix in Chapter 7

Sv Spectral velocity 

SI Spectrum Intensity 

tdb Bracketed duration of strong motion 

tdi Intensity-based duration of strong motion defined by Trifunac and Brady 

(1975) 

Tb Natural period of the base isolation system

TG Predominant period of the earthquake ground motion

Tn Fundamental period of the structure 

Tpa Predominant period of earthquake ground motion based on acceleration

response spectra 

Tpe Predominant period of earthquake ground motion based on acceleration

energy spectra

u Relative displacement; deformation

ug Ground displacement 

um Maximum deformation

urel Relative displacement 

uy Yield deformation 

uy1 Yield deformation for the first story

u& Relative velocity of the structure with respect to ground 

ut && Absolute (total) acceleration of the structure with respect to ground 

VE Equivalent velocity 

74
W Weight of the each floor of the structure 

z A non-dimensional hysteretic parameter 

α Post- to pre- yield stiffness ratio

β Parameter of Bouc-Wen model defining the shape of hysteresis loop 

γ Parameter of Bouc-Wen model defining the shape of hysteresis loop 

μ Ductility ratio 

μmax Maximum inter-story ductility ratio

ξ Damping ratio; a mass vector in Chapter 7 

ξ b Damping ratio for base isolation system

ω0 Pre-yield frequency for Bouc-Wen model 

ξ 0 Viscous damping ratio for Bouc-Wen model 

75