Analytical solutions are obtained for thermally induced axisymmetric elastic and elastic-plastic deformations in nonuniform heat-generating composite tubes with fixed ends. Thermoelastic solutions are obtained using four different boundary conditions: (i) free, (ii) radially constrained and free, (iii) free and pressurized, (iv) free and radially constrained. Elastic-plastic solutions are obtained for a composite tube having free inner and radially constrained outer boundaries using Tresca's yield condition and its associated flow rule. The first two stages of elastic-plastic deformations are studied considering nonlinearly hardening, linearly hardening, and perfectly plastic material behavior. The theory developed is illustrated in several numerical examples.