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$$\begin{array}{c} \text{statics}\\ \hline R=-\sum_j P_j\\ M_A=-\sum_j M_j-\sum_j r_j\times P_j\\ \end{array}$$ $$\begin{array}{c} \text{centroids}\\ \hline \bar x=\begin{cases} \frac{\int_A xdA}{\int_A dA}&\text{single}\\ \frac{\sum_j\bar x_j A_j}{\sum_j A_j}&\text{collection}\\ 0&\text{symmetrical}\\ \frac{h}{3}&\text{triangle}\\ \frac{4r}{3\pi}&\text{semicircle}\\ \end{cases} \end{array}$$ $$\begin{array}{c} \text{distributed}\\ \hline F=\int_L fdx\\ M=\int_L r\times fdx\\ r\times F=M\\ r\stackrel{r\perp F}=\frac{M}{F}\\ \end{array}$$ $$\begin{array}{c} \text{conventions}\\ \hline \sigma>0,\ \varepsilon>0\text{ if tensile}\\ \gamma>0\text{ if angle decreases}\\ \end{array}$$ $$\begin{array}{c} \text{traction}\\ \hline t=\sigma+\tau\stackrel{[\cdot]}=\frac{F}{A}\\ t_{av}=\frac{1}{A}\int_A tdA=\frac{P}{A}\\ \end{array}$$ $$\begin{array}{c} \text{skew plane area}\\ \hline A'=A\sec\theta \end{array}$$ $$\begin{array}{c} \text{strain}\\ \hline \varepsilon=\frac{dL'-dL}{dL}\\ \delta=\int_L\varepsilon dL\\ \gamma=-\Delta\theta=\frac{\pi}{2}-\arccos\frac{x_i'\cdot x_j'}{|x_i'||x_j'|} \end{array}$$ $$\begin{array}{c} \text{isotropic linear elasticity}\\ \hline \varepsilon_i=\frac{1}{E}(\sigma_i-\nu\sum_{j\ne i}\sigma_j)\\ \gamma_{ij}=\frac{\tau_{ij}}{G}\\ [E]=\frac{F}{A}\text{ (elastic modulus)}\\ [G]=\frac{F}{A}\text{ (shear modulus)}\\ [\nu]=1\text{ (Poisson's ratio)}\\ \hline \varepsilon_{ij}=\frac{1}{E}[(1+\nu)\sigma_{ij}-\nu\delta_{ij}\sigma_{kk}] \end{array}$$