In this paper, numerical formulations are presented; these permit full thermal simulation of an arbitrary plane axisymmetric residual stress field encompassing hoop, radial, and axial stresses. Earlier formulations were based upon the determination of a temperature profile within the tube that could only replicate radial and hoop stresses; in general, axial stresses were incorrect. This new thermal simulation provides all three stresses and is achieved by incorporating orthotropic coefficients of thermal expansion that themselves vary with radius. Results are generally highly accurate. Crucial near-bore hoop and axial stresses can be replicated within 1%. Near-bore behavior is discussed in detail. These formulations will permit subsequent determination of stress intensity factors (SIF) for arbitrarily orientated cracks within pressure vessels in the presence of pre-existing residual stresses. Note that these thermal solutions mimic known, residual stress profiles; they do not predict residual stress profiles.