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State-of-the-Art Review of Energy-Based Seismic Design Methods


This paper presents a comprehensive state-of-the-art review of the research carried out on the energy-based structural seismic design methods. Since earthquake exerts energy to the structure, it is realistic to use the energy as the main design criteria of the structure. The energy-based seismic design method is based on the concept of energy balance in the structures, which states that the earthquake input energy to the structure must be less than its capacity to dissipate the energy; otherwise, local or global damage will occur. Although the energy-based design method has received increasing attention in recent years, it has not yet become an applicable engineering design procedure. This could be due to the lack of specified and reliable criteria for this method. This paper is intended to provide the reader with a thorough review of energy-based seismic design methods, highlighting the unresolved issues and identifying the gaps that will require attention in future research.

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\(a_{rms}\) :

Root-mean-square of acceleration in g

\(AE_{I}\) :

Seismic hazard energy factor

\(A_{pi}\) :

A non-dimensional index of the damage in the i-th story

\(c\) :

Viscous damping coefficient

\(c_{a}\) :

Ratio of spectral elastic response acceleration to PGA in the short (acceleration-controlled) period range

\(c_{v}\) :

Ratio of spectral elastic response velocity to PGV in the medium (velocity-controlled) period range

\(c_{E}\) :

Coefficient that depends on the damping model

\(c_{H}\) :

Coefficient that depends on the hysteretic model

\(CAV\) :

Cumulative absolute velocity

\(d_{n}\) :

Displacement at story n

\(D_{e}\) :

Maximum displacement of the corresponding elastic system

\(D_{y}\) :

Yield displacement

\(D_{max}\) :

Maximum displacement

\(E_{d}\) :

Damping energy

\(E_{D}\) :

The input energy to a single-degree-of-freedom (SDOF) system that will cause structural failure

\(\it {\text{E}}_{{\text{e}}}\) :

Elastic energy

\(E_{ep}\) :

Energy required to push a SDOF system with elasto-perfectly plastic behavior up to the maximum target deformation

\(E_{h}\) :

Hysteretic energy

\(E_{hi}\) :

Energy absorbed in the i-th story

\(E_{h_{r}}\) :

Hysteretic energy in the \(r\)-th mode of the MDOF system

\(E_{h}^{j} \left( {t_{1} } \right)\) :

Hysteretic energy demand in the \(j\)th story of the structure

\(\it \it E_{h\;\;\;\;\;\;_{r}}^{SDOF}\) :

Hysteretic energy in the SDOF system corresponding to \(r\)-th mode

\(E_{h}^{0.1g}\) :

Hysteretic energy corresponding to PGA = 0.1 g

\(E_{I}\) :

Input energy

\(E_{Ia}\) :

Absolute input energy

\(E_{Ir}\) :

Relative input energy

\(E_{I_{r}}\) :

Relative input energy in the \(r\)th mode of MDOF system

\(E_{I\;\;\;\;\;\;_{r}}^{ESDOF}\) :

Relative input energy in the ESDOF system corresponding to \(r\)-th mode of MDOF system

\(E_{I}^{0.1g}\) :

Input energy corresponding to PGA = 0.1 g

\(E_{I}^{*}\) :

Modified input energy

\(E_{k}\) :

Kinetic energy

\(E_{ka}\) :

Absolute kinetic energy

\(E_{kr}\) :

Relative kinetic energy

\(E_{p}\) :

Plastic deformation energy

\(E_{pi}\) :

Plastic deformation energyin i-th story

\(E_{s}\) :

Strain energy

\(E_{se}\) :

Elastic strain energy

\(f_{s}\) :

Damping, restoring force

\(f_{E}\) :

Elastic design input energy spectral shape

\(f_{T}\) :

A period factor

\(\overline{f}\) :

A factor accounting for the ductility and ground motion characteristics

\(F_{i}\) :

Equivalent inertia force at level \(i\)

\(F_{n}\) :

Modal force acting at story n

\(g\) :

Gravitational acceleration

\(h_{i}\) :

Height of beam level \(i\) from the ground

\(I_{A}\) :

Arias intensity index

\(I_{c}\) :

Earthquake intensity index

\(I_{e}\) :

Cumulative damage potential

\(k_{i}\) :

Stiffness of the i-th story

\(m\) :


\(M_{n}^{*}\) :

Effective modal mass of \(n\)-th mode

\(M_{pbi}\) :

Plastic moment of the beam at level \(i\)

\(M_{pc}\) :

Plastic moment of the columns at the base of the structure

\(n_{eq}\) :

Equivalent number of cycles

\(NE\) :

Normalized input energy

\(NE_{I,max,\zeta ,\mu }\) :

Normalized input energy in a structure with desired damping and ductility

\(NE_{I,max,\zeta = 0.02,\mu = 1}\) :

Normalized input energy in an elastic SDOF system with a damping ratio of 2%

\(p_{i}\) :

A parameter that indicates the amount of deviation of the shear of the i-th story from the optimal shear distribution

\(PGV\) :

Peak ground velocity

\(PGA\) :

Peak ground acceleration

\(R_{y}\) :

Yield strength reduction factor

\(S\) :

Soil profile coefficient

\(S_{v}\) :

Spectral pseudo-velocity response

\(S_{v_{max}}\) :

Maximum spectral pseudo-velocity response

\(S_{a}\) :

Spectral pseudo-acceleration response

\(S_{{a@S_{vmax} }}\) :

Spectral pseudo-acceleration response corresponding to \(S_{v_{max}}\)

\(t\) :


\(t_{d}\) :

Duration of the earthquake

\(t_{e}\) :

Effective duration

\(t_{f}\) :

Duration of the earthquake based on the Trifunac–Brady method

\(T\) :

Natural period of the structure

\(T_{c}\) :

Characteristic period of the ground motion

\(T_{eq}\) :

Equivalent period

\(T_{g}\) :

Predominant period of the earthquake

\(u\) :

Relative displacement of system

\(\dot{u}\) :

Relative velocity of system

\(\ddot{u}\) :

Relative acceleration of system

\(u_{g}\) :

Base/ground displacement

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

Base/ground acceleration

\(u_{t}\) :

Absolute (or total) displacement of system

\(V_{by}\) :

Yield base shear

\(V_{e}\) :

Maximum strength of the corresponding elastic system

\(V_{I}\) :

Input energy equivalent velocity

\(V_{Ia}\) :

Absolute input energy equivalent velocity

\(V_{Ir}\) :

Relative input energy equivalent velocity

\(V_{h}\) :

Hysteretic energy equivalent velocity

\(V_{y}\) :

Yield strength

\(w_{i}\) :

Weight of \(i\)th story

\(\alpha\) :

Ratio of damping energy to input energy

\(\alpha_{opt,i}\) :

The optimum yield shear coefficient of the i-th story

\(\overline{{\alpha_{i} }}\) :

The optimum yield-shear coefficient distribution of the i-th story

\(\overline{{{}_{r}^{ } \alpha_{i} }}\) :

Story-shear coefficient distribution

\(\gamma\) :

Energy factor

\(\gamma_{f}\) :

A non-dimensional parameter which controls the low-cycle fatigue effects

\(\zeta\) :

Damping ratio of the structure

\(\zeta_{eq}\) :

Equivalent damping

\(\eta_{p}\) :

Plastic energy modification factor

\(\theta_{p}\) :

The‏ ‏cumulative plastic rotation capacity of the beam

\(\lambda\) :

A parameter that characterizes the spectral shape of the input energy spectrum

\(\lambda_{d}\) :

Modification factor of input energy due to the damping

\(\mu\) :

Displacement ductility factor

\(\mu_{cum}\) :

Cumulative ductility

\(\tau\) :

A measure of the number of cycles of ground motion

\({\varphi }_{n,r}\) :

The i-th story value of the r-th mode shape

\(\psi\) :

Amplification factor of the seismic input energy equivalent velocity

\(\omega\) :

Angular frequency of oscillator

\(\Omega_{v}\) :

Amplification factor of the input energy spectrum

\(\Omega_{v}^{*}\) :

Peak amplification factor for the input energy spectrum


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Correspondence to Sadegh Garivani.

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Gholami, N., Garivani, S. & Askariani, S.S. State-of-the-Art Review of Energy-Based Seismic Design Methods. Arch Computat Methods Eng 29, 1965–1996 (2022).

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