Structural & machine design extras

1. Column buckling (Euler & Johnson)
2. Mohr's circle for plane stress
3. Stress-strain curve
4. Section properties
5. Multi-bolt pattern (Voici)
6. Lifting lug analysis
7. Spring design
8. Gear train ratio & torque
9. Belt drive sizing
10. Bearing life (L10)
11. Fillet weld strength
12. Thin-walled pressure vessel
13. Pipe pressure drop (Voici)
14. Fatigue crack growth (Voici)
15. Process capability C_p/C_pk
16. Electric motor sizing

$$P_{cr}^{\text{Euler}}=\frac{\pi^2 EI}{(KL)^2},\quad \left(\frac{KL}{r}\right)_{c}=\sqrt{\frac{2\pi^2 E}{S_y}},\quad P_{cr}^{\text{Johnson}}=S_y A\left[1-\frac{S_y}{4\pi^2 E}\!\left(\frac{KL}{r}\right)^2\right]$$

$$\sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2},\quad \tau_{max}=R,\quad \theta_p=\frac{1}{2}\tan^{-1}\!\left(\frac{2\tau_{xy}}{\sigma_x-\sigma_y}\right)$$

$$\sigma=\begin{cases}E\varepsilon&\varepsilon\le\varepsilon_y\\\sigma_y+K(\varepsilon-\varepsilon_y)^n&\varepsilon>\varepsilon_y\end{cases}$$

With resilience $U_r=\sigma_y^2/(2E)$ and approximate toughness $U_t\approx0.5(\sigma_y+\sigma_u)\varepsilon_f$.

$$I=\int y^2\,dA,\quad S=\frac{I}{c},\quad r_g=\sqrt{\frac{I}{A}}$$

$$F_{\text{direct}}=\frac{P}{n},\quad F_{\text{torsion},i}=\frac{Mr_i}{\sum r_i^2}$$

Interactive bolt-pattern analysis with force-vector plot. Requires Voici server.

$$\sigma_t=\frac{P}{(w-d)t},\quad \tau_{so}=\frac{P}{2at},\quad \sigma_b=\frac{P}{dt}$$

$$k=\frac{Gd^4}{8D^3N},\quad K_w=\frac{4C-1}{4C-4}+\frac{0.615}{C},\quad \tau=\frac{8FDK_w}{\pi d^3},\quad C=\frac{D}{d}$$

$$\frac{N_1}{N_2}=\frac{\omega_2}{\omega_1}=\frac{T_1}{T_2\;\eta}$$

$$L=2C+\frac{\pi}{2}(D_1+D_2)+\frac{(D_2-D_1)^2}{4C},\quad \theta_1=\pi-2\sin^{-1}\!\left(\frac{D_2-D_1}{2C}\right)$$

$$L_{10}=\left(\frac{C}{P}\right)^p,\quad L_{10h}=\frac{10^6L_{10}}{60n}$$

p = 3 for ball bearings, p = 10/3 for roller bearings.

$$\tau=\frac{P}{0.707wL},\quad w_{\text{req}}=\frac{P}{0.707L\tau_{\text{allow}}}$$

$$\sigma_h=\frac{pr}{t},\quad \sigma_l=\frac{pr}{2t}\ \text{(cyl)},\quad \sigma_{\text{sph}}=\frac{pr}{2t}$$

$$\frac{1}{\sqrt{f}}=-2\log_{10}\!\left(\frac{\varepsilon/D}{3.7}+\frac{2.51}{Re\sqrt{f}}\right),\quad \Delta p=f\frac{L}{D}\frac{\rho v^2}{2}$$

Solves Colebrook-White iteratively and plots operating point on Moody chart. Requires Voici server.

$$\frac{da}{dN}=C(\Delta K)^m,\quad \Delta K=\Delta\sigma\sqrt{\pi a}\,Y$$

Numeric integration of crack length vs cycles to failure. Requires Voici server.

$$C_p=\frac{USL-LSL}{6\sigma},\quad C_{pk}=\min\!\left(\frac{USL-\mu}{3\sigma},\frac{\mu-LSL}{3\sigma}\right)$$

$$P=T\omega=\frac{T\,n}{9550}\ \text{(kW)},\quad T=\frac{9.55\,P}{n}$$