INTRODUCTION
Housner was first who used concept of input energy as seismic design criteria
(Housner, 1956). He presented his pioneering work in
the 1st WCEE in 1956. Three main conclusions of his paper are of special concern
in this study:
• 
The seismic energy input to a SDOF structure with specified
damping, if it seen from spectral or average standpoint, is basically constant
and independent from its period, especially for low damping ratios 
• 
Seismic design of structures may by thought as satisfying the following
inequality: Energy absorption capacity > seismic input energy. On the
other hand, the amount of energy input to an elastic system is the upper
bound of energy input to hysteretic systems with the same liner properties.
So, seismic design of structures dose not means providing too strong elements
with the capability of converting kinetic energy of structure to elastic
strain energy and instead it is enough to supply sufficient capacity of
energy absorption via plastic deformations in structural elements 
• 
Seismic energy input to MDOF systems basically depends to their total
mass and therefore it is equals to energy input to an equivalent SDOF with
the same mass and main period of vibration 
Based on Housner's work, Akiyama published his very important book Earthquake
Resistant Limit State Design for Buildings in 1985. He expanded Housner's assumptions
and showed their limitations and strong points (Akiyama, 1985).
He developed input energy spectrum for different site soils. Those spectrums
are basically constant with respect to the period of vibration except for periods
smaller than predominant periods for the ground. As Housner, Akiyama tried to
simplify seismic design of structures by supposing and demonstrating that input
energy to structures is related mainly to earthquake excitation but scarcely
to structural features. Most of researchers adapted this assumption and equation
proposed by Kuwamura and Galambos (1989), Fajfar
et al. (1989), Uang and Bertero (1990) Kuwamura
et al. (1994), for establishing the earthquake input energy are based
on the ground motion characteristics only.
Parallel to these works in estimating the energy demand, other researchers
focused on the mechanism of dissipation of the input energy in structural elements
by the hysteretic action. Park and Ang (1985) related
their damage index to the energy dissipation via hysteretic loops. This means
that reduction in the hysteretic dissipation of the input energy which is a
fraction of the total input energy, reduces the structural damage. So it arises
an important question:
Is it possible to minimize the seismic input energy to structures by a specific design pattern?
To answer the question, classical approach to the input energy demand problem must be reexamined the task which is the main purpose of this study.
Various aspects of seismic input energy and its calculation, like absolute
and relative energies, time interval for integration of related equations and
so forth, have been discussed in the literature (Kuwamura
and Galambos, 1989; Uang and Bertero, 1990; Khashaee,
2003). In this study some basic assumptions and definitions which are widely
used in the literature are adopted as follow:
• 
Relative, rather than absolute input energy is studied 
• 
Input energy is defined as the energy induced to the structure by strong
ground motion from beginning (t = 0) of motion to the end of motion (t =
t0). Note that definitions of beginning and ending moments of ground motion
are not unique in the literature, but this is not important in this study.
In this study beginning and end of motion are assumed to be coincide to
the beginning and end of record. Also it is demonstrated that the maximum
input energy may be attained not necessarily at the end of motion (Uang
and Bertero, 1990). However, as mentioned earlier, the input energy
of the system at the ending moment of ground motion is considered as seismic
input energy. So the seismic input energy to a SDOF system with mass m,
frequency ω and damping ratio ζ is defined mathematically as: 
where,
and are,
respectively ground acceleration and system velocity. For a system with unit
mass Eq. 1 can be written as:
It is helpful to use an equivalent velocity V_{E}, defined based on
the input energy, as (Housner, 1956):
where, E_{1 }is input energy to the SDOF.
The damping matrix is diagonal and c_{ii} = c ≈ const.
By using the abovementioned assumptions, in the following sections of the paper at first some input energy spectra are obtained and then possibility of existence of optimal stiffness distribution demonstrated mathematically.
INPUT ENERGY SPECTRA
Based on definition in of the input energy, given in the previous section,
some input energy spectra are obtained using ten typical earthquake ground motion
records shown in Table 1. All of these records have been extracted
from peer strong motion database.
All records have been normalized to 1.0 g. Figure 13
show equivalent velocity V_{E} spectra versus period of vibration respectively
for ζ = 0, 0.05 and 0.10. Design Input Energy Spectrum (DIES), proposed
by Akiyama (1985), is also shown in the Fig.
1. It should be mentioned that the shown DIES values are for very stiff
site soil and damping ratio of 10%.
Two important conclusions may be taken from these Fig. 13.
First is the low decay rate of average spectrum with increase in the period
of vibration, especially in the practical range of periods of high rise buildings,
say 0.8 to 5 seconds. For example, it can be seen in Fig. 3
that the amount of input energy at T = 5 sec is half of that at T = 0.9, but
typically, spectral pseudo acceleration at T = 5 sec is less than one fifth
of respect value at T = 0.9 (considering UBC97 design spectrum). Second important
observation is very low sensitivity of the input energy to the damping of structures.
Table 1: 
List of records used to obtain input energy spectra 


Fig. 1: 
Input energy spectra for ζ = 0%. Input energy spectra
damping ratio = 0.00 

Fig. 2: 
Input energy spectra for ζ = 5%. Input energy spectra
damping ratio = 0.05 

Fig. 3: 
Input Energy spectra for ζ = 10%.Input energy spectra
damping ratio = 0.10 
Thus, the DIES can be assumed to be essentially constant over a wide range
of periods, which is in agreement with Housner's pioneering statements. However,
it is important to note that the matter is looked at from the design spectrum
point of view. This means that the results are valid for average values obtained
from past earthquakes, not for an individual record.

Fig. 4: 
Input energy spectra of the considered earthquakes for ζ=10.
Input energy spectra damping ratio = 0.10. (a) ChiChi, (b) Duzce, (c) ClCentro,
(d) Gazli, (e) Kobe, (f) Landers, (g) Lomaprieta, (h) Manjil, (i) Northridge
and (j) Tabass 
Also, it should be noted that to obtain the input energy spectra in this study,
records were selected regardless of their site specification features, which
does not affect the outcome of the study, but some discrepancies may be seen
between DIES and average of spectra (Climent et al.,
2002).
Individual input energy spectra and pseudo velocity for each record are shown
respectively in Fig. 4 and 5 for damping
ratio of 10%. As seen in these two figures, there is a strong correlation between
the input energy and pseudo velocity spectrum. Therefore, pseudo velocity is
an important index of seismic input energy as stated by Housner
(1956).
Comparing Fig. 4aj and 5aj
one can see that the general pattern of the input energy spectrum for each record
is very similar to its corresponding pseudo velocity spectrum.

Fig. 5: 
Pseudo velocity spectra for ζ = 10%. Pseudo velocity
spectra damping ratio = 0.10. (a) ChiChi, (b) Duzce, (c) ClCentro, (d) Gazli,
(e) Kobe, (f) Landers, (g) Lomaprieta, (h) Manjil, (i) Northridge, (j) Tabas 
EQUATION OF MOTION AND INPUT ENERGY TO MDOF SYSTEMS
A schematic illustration of the simplified model of multistory buildings, considered in this study, is shown in Fig. 6.
The equation of motion of the system shown in the Fig. 6 can be written as:
or:
Where:
It is obvious that the input energy to this system can be written as:
Thus:
Now by decupling Eq. 2 results in:
Equation 5 can be interpreted as equation of motion of a
SDOF system with unit mass subjected to ground acceleration ,
magnified by .
It is evident that magnifying the excitation by leads
to magnification of the input energy by ,
thus considering Eq. 2 the input energy to the system
shown by Eq. 5 may be written as:
By comparing Eq. 4 and 6 the following
Equation can be written:
But, as indicated previously, in a wide range of relatively long to very long
periods has
no notable variation and may be taken as constant and thus Eq.
7 can be written as:
Based on Eq. 8 it can be claimed that, considering E as a constant value, the summation term in the last this Equation must be a constant. In fact, it can be said that as the mode shapes of any system are bases of a vector space V, each vector, including {1}, in this space can be written as a linear combination of bases, which is:
Thus, if {v} = {1} then [Φ]{a} = {1} and one can write:
But, [Φ][Φ]^{T}[M] in the right hand side of Eq. 10 must be a unit matrix because [Φ]^{T}[M][Φ] = [I] and by premultiplication of both sides by [Φ] one can obtain the desired result. Thus, Eq. 10 may be rewritten as:
and finally, Eq. 8 can be written as:
This means that the seismic input energy to a MDOF system is the same as input
energy to a SDOF system with same mass, main frequency and damping, provided
that the following conditions are met:
• 
Input energy is calculated at a specified instant of a
record for all modes, say at the end of recor

• 
Input energy spectra are constant all over the wide range of periods 
As it can be seen from Fig. 13, constancy of input energy spectra is a simplifying assumption which apparently is not fully in agreement with reality, particularly in the rage of short to medium periods. In fact, the input energy as expressed by Eq. 7 depends on the shape of spectra and therefore, on the structural features. On the other hand, calculation of the input energy at a specified instant of a record, say at its end, implies that unlike base shear and other desired quantities, the input energy is maximized simultaneously in all modes and therefore there is no need to use modal combination methods such as SRSS and CQC. This fact simplifies the problem and in conjugation with the inconstant input energy spectrum, demonstrates possibility of existence of an optimal distribution of stiffness in high rise buildings to minimize the seismic input energy to the structure.
CONCLUSIONS
Followings are the main conclusions of this study (1) Constancy of input energy spectrum is a simplified assumption which is not in agreement with reality, particularly in the rage of short to moderate periods, which correspond to majority of short to midrise buildings, (2) Equality of the amount of energy, input to a MDOF system, with that of a SDOF system is limited to the validity of constancy of input energy spectrum and (3) It is conceptually possible to find an optimal distribution of stiffness over the height of a multistory building to minimize the seismic input energy to the structure. However, it should be pointed out that the hysteretic energy dissipated by the structure is an important damage index and that; the hysteretic energy to be dissipated by the system can be expressed as a fraction of the total input energy.