The traditional Lekhnitskii 3D elastic theory can obtain the exact stress and strain response of the composite cylinder under the axially symmetric loadings. However, when the cylinder contains 0° winding layers or isotropic layers, some calculation parameters corresponding to the two kinds of special layers will become singular. By analyzing the continuity of singular parameters and calculating their limits, the problem of singular parameters can be effectively solved and the Lekhnitskii theory can be extended to composite thick-walled cylinders with arbitrary winding angles. Furthermore, based on the Chang-Chang strength criterion of composite layers and the ideal elastic-plastic assumption of isotropic metal layers, the theoretical calculation method of first layer failure strength and the corresponding failure modes of the composite cylinders subjected to axisymmetric loadings was proposed. Then, the composite cylinder with metal liner was fabricated by fiber winding process. The experimental blasting pressure of the cylinder was 52.8 MPa with a major failure mode of fiber tensile fracture. Based on the proposed theoretical method, the cylinder's first layer failure strength of internal pressure was 44.1 MPa, and the initial failure mode was also fiber tensile fracture. The predicted result of the first layer failure strength was lower than the experimental results. The result proved the safety and economy of the proposed theoretical method for the initial design of the composite cylinder.