THE kinetic heating associated with supersonic flight produces temperature gradients within the aircraft structure. These in their turn are responsible for so-called 'thermal stresses' in the components. The calculation of these effects falls into two stages. The first stage consists in the application of the theory of heat transfer to obtain the history of the temperature distribution in the structure. The second stage uses this data to obtain distributions of stress within the structure, resulting from these imposed temperature gradients and proceeds to assess their influence on strength and stiffness. The present paper is concerned entirely with this second stage of the problem and derives basic formulae for the analysis of beam-like structures and components. The results can be applied to wings, fuselages, etc., on the one hand, and to linear reinforcing members like stringers and longerons on the other, in the same way as the usual theories of bending and torsion are applied in the isothermal case. The formulae obtained in this paper represent a generalization of the so-called engineering theory of bending and of the Wagner-Kappus torsion theory to include the effects of non-uniform temperature distribution. Kinematically, allowance is made for overall longitudinal extension, for curvature in two principal planes, for twist and for cross-sectional warping of the kind occurring in Saint Venant's torsion theory. Relationships between end load, bending moments and torques on the one hand and the kinematic parameters on the other are obtained, in a manner modelled on that of Ref. (1), by means of a 'Principle of Stationary Free Energy' established by the present writer in Ref. (2). These results, when combined with the well-known equilibrium equations for bending and torsion, constitute a complete theory of the problem under consideration. Applications to problems of stress analysis are indicated.