The Spectra of Relative Input Energy per Unit Mass of Structure for Iranian Earthquakes

  • Reza VahdaniEmail author
  • Mohsen Gerami
  • Mohamad Ali Vaseghi-Nia
Research paper


Energy criterion is a simple and scalar quantity, so it has been employed by many researchers for the assessment of seismic behavior. Production and presentation of input energy spectra are effective steps for the employment of energy criterion in the seismic design of structures. Ninety-two pairs of horizontal components of Iranian earthquakes were used for this research. These records were divided into near-field and far-field, and in each field, soils categorized to types 1–3. Using nonlinear dynamic analysis, 92 inelastic spectra of relative input energy per unit mass were generated for a damping ratio of 5% and a ductility factor of 3. Then, for each category of records, a combined spectrum was produced at design level corresponding to 10% risk in 50 years. The evaluation of combined spectra led to the conclusion that the average value of combined spectrum in near-field is greater than that in far-field. In addition, the average value of combined spectrum is greater for softer soils. The corresponding period to the peak of combined spectrum in near-field is longer than that in far-field. The effect of soil type in near-field is more than that in far-field. For each field and each type of soil, a relation and its parameters have been proposed for the inelastic spectrum of relative input energy per unit mass.


Inelastic spectrum Soil type Near-field Far-field Seismic design Combined spectrum 

List of symbols


Total mass of structure


Viscous damping coefficient of structure


Restoring force of structure


Fundamental period of structure


Corresponding period to the peak of combined spectrum


Displacement of mass relative to the ground


Displacement of the ground during an earthquake


Total displacement (absolute displacement) of mass

\({\dot {u}_{\text{g}}}\)

Velocity of mass relative to the ground

\(\dot {u}\)

Velocity of the ground during an earthquake

\({\dot {u}_{_{{\text{t}}}}}\)

Total velocity (absolute velocity) of mass

\(\ddot {u}\)

Acceleration of mass relative to the ground

\({\ddot {u}_{\text{g}}}\)

Acceleration of the ground during an earthquake

\({\ddot {u}_{\text{t}}}\)

Total acceleration (absolute acceleration) of mass


Relative input energy


Relative kinematic energy


Damping energy


Strain absorbed energy


Damping ratio of structure


Displacement ductility factor of structure


Maximum displacement of mass


Yielding displacement of structure

γ, β

Coefficients of Newmark method


Relative input energy per unit mass of structure


Maximum value of \(\frac{{{E_{{\text{ri}}}}}}{m}\) in proposed spectra

α1, α2

Coefficients of the proposed spectra of \(\frac{{{E_{{\text{ri}}}}}}{m}\)


Corresponding period to the beginning of constant segment in the proposed spectra of \(\frac{{{E_{{\text{ri}}}}}}{m}\)


Corresponding period to the ending of constant segment in the proposed spectra of \(\frac{{{E_{{\text{ri}}}}}}{m}\)


Standard deviation


Mean spectrum

Index x

Related to x-direction

Index y

Related to y-direction


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Copyright information

© Iran University of Science and Technology 2018

Authors and Affiliations

  • Reza Vahdani
    • 1
    Email author
  • Mohsen Gerami
    • 1
  • Mohamad Ali Vaseghi-Nia
    • 1
  1. 1.Department of Earthquake EngineeringSemnan UniversitySemnanIran

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