Formulas to obtain mechanical properties of elastomeric bearing
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 | ${{u}_{c}}$ | $\frac{{{F}_{c}}}{{{K}_{v0}}}$ |