SullivanMuse.github.io-Old/structures.html at main · SullivanMuse/SullivanMuse.github.io-Old · GitHub

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      $$\begin{array}{c}
      \text{statics}\\
      \hline
      R=-\sum_j P_j\\
      M_A=-\sum_j M_j-\sum_j r_j\times P_j\\
      \end{array}$$

      $$\begin{array}{c}
      \text{centroids}\\
      \hline
      \bar x=\begin{cases}
      \frac{\int_A xdA}{\int_A dA}&\text{single}\\
      \frac{\sum_j\bar x_j A_j}{\sum_j A_j}&\text{collection}\\
      0&\text{symmetrical}\\
      \frac{h}{3}&\text{triangle}\\
      \frac{4r}{3\pi}&\text{semicircle}\\
      \end{cases}
      \end{array}$$

      $$\begin{array}{c}
      \text{distributed}\\
      \hline
      F=\int_L fdx\\
      M=\int_L r\times fdx\\
      r\times F=M\\
      r\stackrel{r\perp F}=\frac{M}{F}\\
      \end{array}$$

      $$\begin{array}{c}
      \text{conventions}\\
      \hline
      \sigma>0,\ \varepsilon>0\text{ if tensile}\\
      \gamma>0\text{ if angle decreases}\\
      \end{array}$$

      $$\begin{array}{c}
      \text{traction}\\
      \hline
      t=\sigma+\tau\stackrel{[\cdot]}=\frac{F}{A}\\
      t_{av}=\frac{1}{A}\int_A tdA=\frac{P}{A}\\
      \end{array}$$

      $$\begin{array}{c}
      \text{skew plane area}\\
      \hline
      A'=A\sec\theta
      \end{array}$$

      $$\begin{array}{c}
      \text{strain}\\
      \hline
      \varepsilon=\frac{dL'-dL}{dL}\\
      \delta=\int_L\varepsilon dL\\
      \gamma=-\Delta\theta=\frac{\pi}{2}-\arccos\frac{x_i'\cdot x_j'}{|x_i'||x_j'|}
      \end{array}$$

      $$\begin{array}{c}
      \text{isotropic linear elasticity}\\
      \hline
      \varepsilon_i=\frac{1}{E}(\sigma_i-\nu\sum_{j\ne i}\sigma_j)\\
      \gamma_{ij}=\frac{\tau_{ij}}{G}\\
      [E]=\frac{F}{A}\text{ (elastic modulus)}\\
      [G]=\frac{F}{A}\text{ (shear modulus)}\\
      [\nu]=1\text{ (Poisson's ratio)}\\
      \hline
      \varepsilon_{ij}=\frac{1}{E}[(1+\nu)\sigma_{ij}-\nu\delta_{ij}\sigma_{kk}]
      \end{array}$$
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