Appendix 1: Formulas for Calculation of Crack Magnitude from Crack Depth
$$ \gamma ={E}_0A/T=2\pi \left(1-{\nu}_0^2\right){hf}_u(z),z=a/h; $$
$$ \begin{array}{l}{f}_u(z)={z}^2\Big(0.6272-0.17248z+5.92134{z}^2-10.7054{z}^3+31.5685{z}^4-67.47{z}^5+\\ {}\kern4.919997em +139.123{z}^6-146.682{z}^7+92.3552{z}^8\Big);\end{array} $$
$$ \theta = EI/R=6\pi \left(1-{\nu}_0^2\right){hf}_w(z); $$
$$ \begin{array}{l}{f}_w(z)={z}^2\Big(0.6272-1.04533z+4.5948{z}^2-9.9736{z}^3+20.2948{z}^4-33.0351{z}^5+\\ {}\kern4.919997em +47.1063{z}^6-40.7556{z}^7+19.6{z}^8\Big).\end{array} $$
Appendix 2: Shape Function for Bar and Beam Elements
$$ {h}_1\left(\alpha, \ell, x\right)=\left[{\varphi}_2\left(\alpha, \ell \right){\varphi}_1\left(\alpha, x\right)-{\varphi}_1\left(\alpha, \ell \right){\varphi}_2\left(\alpha, x\right)\right]/d;{h}_2\left(\alpha, \ell, x\right)=\left[{\varphi}_1\left(\alpha, 0\right){\varphi}_2\left(\alpha, x\right)-{\varphi}_2\left(\alpha, 0\right){\varphi}_1\left(\alpha, x\right)\right]/d; $$
$$ d\left(\alpha, \ell \right)={\varphi}_1\left(\alpha, 0\right){\varphi}_2\left(\alpha, \ell \right)-{\varphi}_1\left(\alpha, \ell \right){\varphi}_2\left(\alpha, 0\right);{H}_j\left(\beta, \ell, x\right)=\left(1/\Delta \right)\sum_{k=1}^4{\left(-1\right)}^{j+k}{\Delta}_{jk}{\Phi}_k\left(\beta, x\right),j=1,2,3,4. $$
$$ {\Delta}_{11}= \det \left[\begin{array}{ccc} {\Phi}_2^{\prime}\left(\beta, 0\right) & {\Phi}_3^{\prime}\left(\beta, 0\right) & {\Phi}_4^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_2\left(\beta, \ell \right) & {\Phi}_3\left(\beta, \ell \right) & {\Phi}_4\left(\beta, \ell \right) \\ {} {\Phi}_2^{\prime}\left(\beta, \ell \right) & {\Phi}_3^{\prime}\left(\beta, \ell \right) & {\Phi}_4^{\prime}\left(\beta, \ell \right) \end{array}\right];{\Delta}_{12}= \det \left[\begin{array}{ccc} {\Phi}_2\left(\beta, 0\right) & {\Phi}_3\left(\beta, 0\right) & {\Phi}_4\left(\beta, 0\right) \\ {} {\Phi}_2\left(\beta, \ell \right) & {\Phi}_3\left({\beta}_2,{\ell}_2\right) & {\Phi}_4\left(\beta, \ell \right) \\ {} {\Phi}_2^{\prime}\left(\beta, \ell \right) & {\Phi}_3^{\prime}\left({\beta}_2,{\ell}_2\right) & {\Phi}_4^{\prime}\left(\beta, \ell \right) \end{array}\right];{\Delta}_{13}= \det \left[\begin{array}{ccc} {\Phi}_2^{\prime}\left(\beta, 0\right) & {\Phi}_3^{\prime}\left(\beta, 0\right) & {\Phi}_4^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_2\left(\beta, 0\right) & {\Phi}_3\left(\beta, 0\right) & {\Phi}_4\left(\beta, 0\right) \\ {} {\Phi}_2^{\prime}\left(\beta, \ell \right) & {\Phi}_3^{\prime}\left(\beta, \ell \right) & {\Phi}_4^{\prime}\left(\beta, \ell \right) \end{array}\right]; $$
$$ {\Delta}_{14}= \det \left[\begin{array}{ccc} {\Phi}_2\left(\beta, 0\right) & {\Phi}_3\left(\beta, 0\right) & {\Phi}_4\left(\beta, 0\right) \\ {} {\Phi}_2^{\prime}\left(\beta, 0\right) & {\Phi}_3^{\prime}\left(\beta, 0\right) & {\Phi}_4^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_2\left(\beta, \ell \right) & {\Phi}_3\left(\beta, \ell \right) & {\Phi}_4\left(\beta, \ell \right) \end{array}\right];{\Delta}_{21}= \det \left[\begin{array}{ccc} {\Phi}_1^{\prime}\left(\beta, 0\right) & {\Phi}_3^{\prime}\left(\beta, 0\right) & {\Phi}_4^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, \ell \right) & {\Phi}_3\left(\beta, \ell \right) & {\Phi}_4\left(\beta, \ell \right) \\ {} {\Phi}_1^{\prime}\left(\beta, \ell \right) & {\Phi}_3^{\prime}\left(\beta, \ell \right) & {\Phi}_4^{\prime}\left(\beta, \ell \right) \end{array}\right];{\Delta}_{22}= \det \left[\begin{array}{ccc} {\Phi}_1\left(\beta, 0\right) & {\Phi}_3\left(\beta, 0\right) & {\Phi}_4\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, \ell \right) & {\Phi}_3\left(\beta, \ell \right) & {\Phi}_4\left(\beta, \ell \right) \\ {} {\Phi}_1^{\prime}\left(\beta, \ell \right) & {\Phi}_3^{\prime}\left(\beta, \ell \right) & {\Phi}_4^{\prime}\left(\beta, \ell \right) \end{array}\right]; $$
$$ {\Delta}_{23}= \det \left[\begin{array}{ccc} {\Phi}_1^{\prime}\left(\beta, 0\right) & {\Phi}_3^{\prime}\left(\beta, 0\right) & {\Phi}_4^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, 0\right) & {\Phi}_3\left(\beta, 0\right) & {\Phi}_4\left(\beta, 0\right) \\ {} {\Phi}_1^{\prime}\left(\beta, \ell \right) & {\Phi}_3^{\prime}\left(\beta, \ell \right) & {\Phi}_4^{\prime}\left(\beta, \ell \right) \end{array}\right];{\Delta}_{24}= \det \left[\begin{array}{ccc} {\Phi}_1\left(\beta, 0\right) & {\Phi}_3\left(\beta, 0\right) & {\Phi}_4\left(\beta, 0\right) \\ {} {\Phi}_1^{\prime}\left(\beta, 0\right) & {\Phi}_3^{\prime}\left(\beta, 0\right) & {\Phi}_4^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, \ell \right) & {\Phi}_3\left(\beta, \ell \right) & {\Phi}_4\left(\beta, \ell \right) \end{array}\right];{\Delta}_{31}= \det \left[\begin{array}{ccc} {\Phi}_1^{\prime}\left(\beta, 0\right) & {\Phi}_2^{\prime}\left(\beta, 0\right) & {\Phi}_4^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, \ell \right) & {\Phi}_2\left(\beta, \ell \right) & {\Phi}_4\left(\beta, \ell \right) \\ {} {\Phi}_1^{\prime}\left(\beta, \ell \right) & {\Phi}_2^{\prime}\left(\beta, \ell \right) & {\Phi}_4^{\prime}\left(\beta, \ell \right) \end{array}\right]; $$
$$ {\Delta}_{32}= \det \left[\begin{array}{ccc} {\Phi}_1\left(\beta, 0\right) & {\Phi}_2\left(\beta, 0\right) & {\Phi}_4\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, \ell \right) & {\Phi}_2\left(\beta, \ell \right) & {\Phi}_4\left(\beta, \ell \right) \\ {} {\Phi}_1^{\prime}\left(\beta, \ell \right) & {\Phi}_2^{\prime}\left(\beta, \ell \right) & {\Phi}_4^{\prime}\left(\beta, \ell \right) \end{array}\right];{\Delta}_{33}= \det \left[\begin{array}{ccc} {\Phi}_1^{\prime}\left(\beta, 0\right) & {\Phi}_2^{\prime}\left(\beta, 0\right) & {\Phi}_4^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, 0\right) & {\Phi}_2\left(\beta, 0\right) & {\Phi}_4\left(\beta, 0\right) \\ {} {\Phi}_1^{\prime}\left(\beta, \ell \right) & {\Phi}_2^{\prime}\left(\beta, \ell \right) & {\Phi}_4^{\prime}\left(\beta, \ell \right) \end{array}\right];{\Delta}_{24}= \det \left[\begin{array}{ccc} {\Phi}_1\left(\beta, 0\right) & {\Phi}_2\left(\beta, 0\right) & {\Phi}_4\left(\beta, 0\right) \\ {} {\Phi}_1^{\prime}\left(\beta, 0\right) & {\Phi}_2^{\prime}\left(\beta, 0\right) & {\Phi}_4^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, \ell \right) & {\Phi}_2\left(\beta, \ell \right) & {\Phi}_4\left(\beta, \ell \right) \end{array}\right]; $$
$$ {\Delta}_{41}= \det \left[\begin{array}{ccc} {\Phi}_1^{\prime}\left(\beta, 0\right) & {\Phi}_2^{\prime}\left(\beta, 0\right) & {\Phi}_3^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, \ell \right) & {\Phi}_2\left(\beta, \ell \right) & {\Phi}_3\left(\beta, \ell \right) \\ {} {\Phi}_1^{\prime}\left(\beta, \ell \right) & {\Phi}_2^{\prime}\left(\beta, \ell \right) & {\Phi}_3^{\prime}\left(\beta, \ell \right) \end{array}\right];{\Delta}_{42}= \det \left[\begin{array}{ccc} {\Phi}_1\left(\beta, 0\right) & {\Phi}_2\left(\beta, 0\right) & {\Phi}_3\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, \ell \right) & {\Phi}_2\left(\beta, \ell \right) & {\Phi}_3\left(\beta, \ell \right) \\ {} {\Phi}_1^{\prime}\left(\beta, \ell \right) & {\Phi}_2^{\prime}\left(\beta, \ell \right) & {\Phi}_3^{\prime}\left(\beta, \ell \right) \end{array}\right];{\Delta}_{43}= \det \left[\begin{array}{ccc} {\Phi}_1^{\prime}\left(\beta, 0\right) & {\Phi}_2^{\prime}\left(\beta, 0\right) & {\Phi}_3^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, 0\right) & {\Phi}_2\left(\beta, 0\right) & {\Phi}_3\left(\beta, 0\right) \\ {} {\Phi}_1^{\prime}\left(\beta, \ell \right) & {\Phi}_2^{\prime}\left(\beta, \ell \right) & {\Phi}_3^{\prime}\left(\beta, \ell \right) \end{array}\right]; $$
$$ {\Delta}_{44}= \det \left[\begin{array}{ccc} {\Phi}_1\left(\beta, 0\right) & {\Phi}_2\left(\beta, 0\right) & {\Phi}_3\left(\beta, 0\right) \\ {} {\Phi}_1^{\prime}\left(\beta, 0\right) & {\Phi}_2^{\prime}\left(\beta, 0\right) & {\Phi}_3^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, \ell \right) & {\Phi}_2\left(\beta, \ell \right) & {\Phi}_3\left(\beta, \ell \right) \end{array}\right];\Delta = \det \left[\begin{array}{cccc} {\Phi}_1\left(\beta, 0\right) & {\Phi}_2\left(\beta, 0\right) & {\Phi}_3\left(\beta, 0\right) & {\Phi}_4\left(\beta, 0\right) \\ {} {\Phi}_1^{\prime}\left(\beta, 0\right) & {\Phi}_2^{\prime}\left(\beta, 0\right) & {\Phi}_3^{\prime}\left(\beta, 0\right) & {\Phi}_4^{\prime}\left(\beta, 0\right) \\ {} {\Phi}_1\left(\beta, \ell \right) & {\Phi}_2\left(\beta, \ell \right) & {\Phi}_3\left(\beta, \ell \right) & {\Phi}_4\left(\beta, \ell \right) \\ {} {\Phi}_1^{\prime}\left(\beta, \ell \right) & {\Phi}_2^{\prime}\left(\beta, \ell \right) & {\Phi}_3^{\prime}\left(\beta, \ell \right) & {\Phi}_4^{\prime}\left(\beta, \ell \right) \end{array}\right]. $$
