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

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## Abstract

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.

## Keywords

Inelastic spectrum Soil type Near-field Far-field Seismic design Combined spectrum## List of symbols

*m*Total mass of structure

*C*Viscous damping coefficient of structure

- \({f_{\text{s}}}\)
Restoring force of structure

*T*Fundamental period of structure

*T*_{p}Corresponding period to the peak of combined spectrum

*u*Displacement of mass relative to the ground

*u*_{g}Displacement of the ground during an earthquake

*u*_{t}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

- E
_{ri} Relative input energy

*E*_{rk}Relative kinematic energy

*E*_{d}Damping energy

*E*_{a}Strain absorbed energy

*ξ*Damping ratio of structure

*µ*Displacement ductility factor of structure

*u*_{m}Maximum displacement of mass

*u*_{y}Yielding displacement of structure

*γ, β*Coefficients of Newmark method

- \(\frac{{{E_{{\text{ri}}}}}}{m}\)
Relative input energy per unit mass of structure

*E*_{0}Maximum value of \(\frac{{{E_{{\text{ri}}}}}}{m}\) in proposed spectra

*α*_{1},*α*_{2}Coefficients of the proposed spectra of \(\frac{{{E_{{\text{ri}}}}}}{m}\)

*T*_{1}Corresponding period to the

*beginning*of constant segment in the proposed spectra of \(\frac{{{E_{{\text{ri}}}}}}{m}\)*T*_{2}Corresponding period to the

*ending*of constant segment in the proposed spectra of \(\frac{{{E_{{\text{ri}}}}}}{m}\)- SD
Standard deviation

- MS
Mean spectrum

- Index
*x* Related to

*x*-direction- Index
*y* Related to

*y*-direction

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