\(I_{\mathrm{A}}=0.00686(D^{4}-d^{4})-\frac{0.0177D^{2}d^{2}(D-d)}{(D+d)}\)

\(Y_{\mathrm{A}}=\frac{2}{3{\pi}}\frac{D^{2}+Dd+d^{2}}{D+d}\)

\(I_{\mathrm{B}}=\frac{Bh^{3}+2b(H-h)^{3}}{12}+Bh\left(Y_{\mathrm{B}}-\frac{h}{2}\right)^{2}+2b(H-h)\left(\frac{H-h}{2}+h-Y_{\mathrm{B}}\right)^{2}\)

\(Y_{\mathrm{B}}=\frac{Bh^{2}+2b(H^{2}-h^{2})}{2[Bh+2b(H-h)]}\)

\(S_{\mathrm{A}}=\frac{{\pi}}{8}(D^{2}-d^{2})\)

\(S_{\mathrm{B}}=2b(H-h)+Bh\)

\(I_{\mathrm{T}}=I_{\mathrm{A}}+S_{\mathrm{A}}(Y_{\mathrm{A}}+Y_{\mathrm{T}})^{2}+I_{\mathrm{B}}+S_{\mathrm{B}}(H-Y_{\mathrm{B}}-Y_{\mathrm{T}})^{2}\)

\(\frac{{\delta}I_{\mathrm{T}}}{{\delta}Y}=2S_{\mathrm{A}}(Y_{\mathrm{A}}+Y_{\mathrm{T}})+2S_{\mathrm{B}}(H-Y_{\mathrm{B}}-Y_{\mathrm{T}})(-1)=0\)

\(Y_{\mathrm{T}}=\frac{S_{\mathrm{B}}(H-Y_{\mathrm{B}})-S_{\mathrm{A}}Y_{\mathrm{A}}}{S_{\mathrm{A}}+S_{\mathrm{B}}}\)

\(Y_{\mathrm{max}}=H-Y_{\mathrm{T}}\)

\({\sigma}_{\mathrm{max}}=\frac{M}{I_{\mathrm{max}}}Y_{\mathrm{max}}\)