research:isolation:properties [Indian Institute of Technology Bombay]

Property	Notation	Formula
Single rubber layer thickness	${{t}_{r}}$
Number of rubber layers	$n$
Steel shim thickness	${{t}_{s}}$
Outer diameter	${{D}_{o}}$
Inner/lead core diameter	${{D}_{i}}$
Rubber cover thickness	${{t}_{c}}$
Shear modulus	$G$
Bulk modulus of rubber	${{K}_{bulk}}$
Yield stress of lead	${{\sigma }_{L}}$
Density of bearing	${{\rho }_{b}}$
Total rubber layer thickness	$t_r$	$n{{t}_{r}}$
Total height	$h$	$n{{t}_{r}}+(n-1){{t}_{s}}$
Bonded rubber area	$A$	$\frac{\pi }{4}\left[ {{\left( {{D}_{o}}+{{t}_{c}} \right)}^{2}}-D_{i}^{2} \right]$
Lead area	${{A}_{L}}$	$\frac{\pi }{4}D_{i}^{2}$
Characteristic strength	${{Q}_{d}}$	${{\sigma }_{L}}{{A}_{L}}$
Yield displacement	$Y$
Shape factor	$S$	$\frac{{{D}_{o}}-{{D}_{i}}}{4{{t}_{r}}}$
Moment of inertia	$I$	$\frac{\pi }{64}\left[ {{\left( {{D}_{o}}+{{t}_{c}} \right)}^{4}}-D_{i}^{4} \right]$
Adjusted moment of inertia	${{I}_{s}}$	$I\frac{h}{{{T}_{r}}}$
Volume of bearing	${{V}_{b}}$	$Ah$
Mass of bearing	${{m}_{b}}$	${{\rho }_{b}}{{V}_{b}}$
Diameter ratio	${{r}_{d}}$	$\frac{{{D}_{o}}}{{{D}_{i}}}$
Central hole factor	$F$	$\frac{{{\left( {{r}_{d}} \right)}^{2}}+1}{{{\left( {{r}_{d}}-1 \right)}^{2}}}+\frac{1+{{r}_{d}}}{\left( 1-{{r}_{d}} \right)\ln \left( {{r}_{d}} \right)}$
Compression modulus	${{E}_{c}}$	${{\left( \frac{1}{6G{{S}^{2}}F}+\frac{4}{3{{K}_{bulk}}} \right)}^{-1}}$
Rotational modulus	${{E}_{r}}$	$\frac{{{E}_{c}}}{3}$
Vertical stiffness	${{K}_{v}}$	$\frac{A{{E}_{c}}}{{{T}_{r}}}$
Horizontal post yield stiffness	${{K}_{d}}$	$\frac{GA}{{{T}_{r}}}$
Horizontal yield strength	${{F}_{Y}}$	${{Q}_{d}}+{{K}_{d}}Y$
Horizontal elastic stiffness	${{K}_{el}}$ ( ${{K}_{1}}$ )	$\frac{{{F}_{Y}}}{Y}$
Stiffness ratio	$\alpha$	$\frac{{{K}_{d}}}{{{K}_{el}}}$ $\left( =\frac{{{K}_{2}}}{{{K}_{1}}} \right)$
Torsional stiffness	${{K}_{t}}$	$\frac{2G{{I}_{s}}}{h}$
Rotational stiffness	${{K}_{r}}$	$\frac{{{E}_{r}}{{I}_{s}}}{h}$
Critical buckling load	${{P}_{cr}}$	$\frac{\pi \sqrt{{{E}_{r}}GI{{A}_{0}}}}{{{T}_{r}}}$
Critical buckling displacement	${{u}_{cr}}$	$\frac{{{P}_{cr}}}{{{K}_{v0}}}$
Cavitation force	${{F}_{c}}$	$3G{{A}_{0}}$
Cavitation displacement	${{u}_{c}}$	$\frac{{{F}_{c}}}{{{K}_{v0}}}$

Formulas to obtain mechanical properties of elastomeric bearing