A finite element formulation for the nonlinear heat conduction and thermoelastic analyses of orthotropic, axisymmetric layered pressure vessels is presented. Nonlinearities include temperature-dependent material properties and stress-dependent layer interface conditions. Solution of the nonlinear heat conduction equations is iteratively obtained using a modified Newton-Raphson scheme. Direct iteration between heat conduction and stress analyses is employed when stress-dependent interface thermal resistance is considered. A modified time integration scheme to reduce oscillatory noise is introduced, and the stability and accuracy of the time integration scheme are discussed. Numerical results for various vessel designs and loadings are presented in Part 2 of the paper.