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| Loaded Beam - Fixed Ends [see Budynas p. 1020 and Machinery's Handbook 29th Ed. p. 264]: |
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\sigma_{a} = \frac{Fab^2y}{Il^2} \sigma_{b} = \frac{2Fa^2b^2y}{Il^3} \sigma_{c} = \frac{Fa^2by}{Il^2}
\Delta_{b} = \frac{Fa^{3}b^{3}}{3EIl^{3}} R_{1} = \frac{Fb^{2}}{l^{3}}(3a+b) R_{2} = \frac{Fa^{2}}{l^{3}}(3b+a)
fos = \frac{\sigma_{yield}}{max(\sigma_{a}, \sigma_{b}, \sigma_{c})}