Types of Stresses to Consider

cylindrical, thin-walled vessel

\begin{aligned} \sigma_{1} =\sigma_{hoop} = \frac{Pr}{t} \end{aligned}

\begin{aligned} \sigma_{2} = \frac{Pr}{2t} = \frac{\sigma_{hoop}}{2}\end{aligned}

circular, thin-walled vessel

\begin{aligned} \sigma_{\circ-1} =\sigma_{\circ-hoop} = \frac{Pr}{2t} \end{aligned}

\begin{aligned} \sigma_{\circ-2} = \frac{Pr}{4t} = \frac{\sigma_{\circ -hoop}}{2}\end{aligned}

σ_normal

\begin{aligned} \sigma_{N} = \frac{P}{A}\end{aligned}

P = axial applied load, or normal force
A = cross sectional area

τ_{at a point}

\begin{aligned} \tau = \frac{VQ}{It}\end{aligned}

V = shear force at the cross-sectional cut, or at point
Q = ȳ’A’, or first moment of area (consider above point, or below)
I = Moment of Inertia of cross-sectional cut
t = thickness at the focal point/point of interest

torsional stress

\begin{aligned} \tau = \frac{Tc}{J} = \frac{Tr}{J}\end{aligned}

T = torque applied (axially)
c = distance from center of shaft to point
r = disance from center of shaft to outer edge of shaft (radius)
J = polar moment of inertia of shaft (is cross-section solid, or hollow?)

torsion equation

\begin{aligned} \frac{\tau_{max}}{r} = \frac{T}{J} = \frac{G\theta}{L}\end{aligned}

τ_max = maximum shear
T = torque applied (axially)
r = disance from center of shaft to outer edge of shaft (radius)

G = shear modulus
J = polar moment of inertia of shaft (is cross-section solid, or hollow?)
θ = angle of twist
L = length of beam

angle of twist

\begin{aligned} \theta = \frac{TL}{JG}\end{aligned}

T = torque applied (axially)
c = distance from center of shaft to point
r = disance from center of shaft to outer edge of shaft (radius)
J = polar moment of inertia of shaft (is cross-section solid, or hollow?)