Precise modelling of thermoelastic cracks remains challenging due to the fact that both of heat flux and stress fields have singularity issue. In the previous studies, the first author proposed different types of symplectic analytical singular element (SASE) for thermal conduction and stress analysis of cracks. It has been demonstrated that these crack-tip elements of which the interior fields are defined by analytical solutions are highly accurate and efficient. However, the thermal mechanical coupling problem of crack cannot be treated with the existing SASEs. The main difficulty is that the analytical solution of the crack problem considering arbitrary temperature distribution is not available. Approximate solution may lead to significant numerical instabilities. Moreover, the construction of a crack-tip singular element for both thermal conduction and stress analysis is complicated and requires more efforts. In this study, the governing symplectic dual equation of thermoelastic crack is restudied. The analytical solution considering arbitrary temperature distribution is obtained in close form which, to the best of the authors' knowledge, has not been found before. Then, the finite element formulation of a new SASE for thermal-mechanical fracture analysis is derived analytically through a variational approach. A two-step analysis procedure is proposed to calculate the mixed mode thermal stress intensity factors (TSIFs)) and the analytis can be done on a fixed finite element mesh. Mesh refinement around the crack tip is unnecessary, and the mixed-mode TSIFs can be solved accurately without any post-processing.