Table 1 Equivalent viscous damping \((\xi _{eq})\) (%) and ductility \((\mu )\) relationships for different structural systems

Precast concrete walls and frames [43]

\(5+\frac{25}{\pi }\left( 1-\frac{1}{\sqrt{\mu }}\right)\)

RC frame [44]

\(\max (4.7, -0.4\mu ^2 + 7.1\mu - 2)\)

RC frame [34]

\(\xi _0+\frac{8}{\pi }\left( 5-\frac{6}{\mu ^{1.8}}+0.2\mu ^{1.8}\right)\)

\(\xi _0\): Initial elastic damping

BRB-RC frame [45]

\(5+\left( \frac{150(\mu -1)(1-0.14)}{(\mu -0.14\mu +0.14\mu ^2)\pi }\right)\)

Steel-braced RC frame [34]

\(\xi _0+\frac{70}{\pi }\left( 1-\frac{43}{\mu ^4}-\frac{4.7}{10^5}\mu ^4\right) ; \mu \ge 3\)

\(\xi _0\): Initial elastic damping

Infilled-RC frame [46]

\(\xi _0+80.4\left( \frac{\mu +0.83}{\mu \pi }\right)\); \(\xi _0+80.4\left( \frac{\mu +0.05}{\mu \pi }\right)\)

\(\xi _0\): Initial elastic damping

Steel members [47]

\(\xi _0+a\left( 1- \frac{1}{\mu ^b}\right) T^c\)

\(\xi _0\): Initial elastic damping, a, b, and c: constants, T: Period

Hybrid-timber [40]

\(5+\frac{62\times \lambda ^{0.4}}{\pi }\left( 1-\frac{1}{\sqrt{\mu }}\right)\)

\(\lambda =H/B\): slenderness ratio, H: height of frame, B: width of the frame

Steel frame with CLT infill [33]

\(\xi _0 + C\left( \frac{\mu -1}{\mu \pi }\right)\)

C: constant