SM Exam Notes

MECHANICS OF MATERIALS
Stress and Strain from tension/compression
(1) Normal Stress [MPa, N/mm²], (2) Normal Strain, (3) Shear Stress (4) Modulus of Elasticity, Young's Modulus, (5) Modulus of Rigidity, Shear Modulus, (6) Bulk Modulus, Hydrostatic Compression Modulus: $\mathrm{(1)}\,\,\sigma= \frac{P}{A}\,\,\,\,\,\,\mathrm{(2)}\,\,\epsilon_{long}= \frac{\Delta L}{L}\,\,\,\,\,\,\mathrm{(3)}\,\,\tau=\frac{V}{A}\,\,\,\,\,\,\mathrm{(4)}\,\,E= \frac{\sigma}{\epsilon_{long}}$ $\mathrm{(5)}\,\,G=\frac{\tau}{\gamma}=\frac{E}{2(1+2\nu)}\,\,\,\,\,\,\mathrm{(6)}\,\,K=\frac{\sigma}{\Delta V/V}=\frac{E}{3(1-2\nu)}$
Stress and Strain from torsion
(1) Torsion Shear Stress with stress concentration K for circular shaft only [MPa, N/mm²] $\mathrm{(1)}\,\,\tau_{max}= K\frac{T r}{J}\,\,\,\,\,\,\mathrm{where}\,\,\,J_{bar}=\frac{\pi D^{4}}{2},\,\,J_{tube}=\frac{\pi}{2}(D^{4}-d^{4})$ (2) Torsion Shear Strain [mm/mm] and (3) Angle of Twist $\mathrm{(2)}\,\,\gamma_{max}=\frac{r \phi}{L}\,\,\,\,\,\mathrm{(3)}\,\,\phi=\sum \frac{TL}{JG}$ (4)Power Transmission [W, Nm/s][hp, 550 ft.lb/s] $\mathrm{(4)}\,\, P = \omega T = (2\pi f) T,\,\,\mathrm{Remember\,\,that:}\,\,f \mathrm{\,\,is\,\,in\,\, Hz=\left(\frac{rev}{s}\right) \left(\frac{2 \pi}{rev}\right)\,\,or\,\,10Hz = 10/s}$ Average shear stress in a thin-walled tube [MPa, N/mm²] $\mathrm{(2)}\,\,\tau_{ave}= \frac{T}{2tA_{m}},\,\,\,\,\mathrm{where}\,\,\tau_{ave} t= \frac{T}{2A_{m}}\,\,\mathrm{is\,\,shear\,\,flow}$
Stress and Strain from bending
$\mathrm{Normal\,\,Stress\,\,[MPa]}\,\,\,\,\sigma= \frac{My}{I}$
Buckling and Columns
A column will buckle about the principal axis of the cross section with the least moment of inertia (the weakest axis) $\mathrm{Critical\,\,Axial\,\,Load\,\,[N,\,\,lb]}\,\,\,\,P_{cr}= \frac{\pi^{2} EI}{(KL)^{2}}$ $\mathrm{Critical\,\,Stress\,\,[MPa]}\,\,\,\,\sigma_{cr}= \frac{\pi^{2} E}{(KL/r)^{2}}$ Where K is the effective-length factor coefficient: 0.5 for fixed ends, 0.7 for pinned and fixed ends, 1.0 for pinned ends, 2.0 for fixed and free ends.
Failure Theories
Coming soon...
Review Prepared by Estiven R. Sierra, 2016

MECHANICS OF MATERIALS