Abstract: The critical buckling loads of viscoelastic laminatedcircular cylindrical shells under axial compression are investigated withinthe theory of classic buckling. Boltzmann hereditary linear constitutiverelationship is used to model the viscoelastic behavior of lamina. Bothgoverning equations of Donnell type and boundary conditions in phase domainare obtained by Laplace transformation. The deflections and in-plane forcefunctions are expressed in series form of separate variables with circumferentialpart in trigonometric functions. The generalized eigenvalue problemin phase domain of determining the critical axial load is studied by means ofthe differential scheme with respect to the axial coordinate. Applying thetheorems of asymptotic value and initial value for Laplace inversetransformation, we obtain, respectively, the formulation of transientelastic critical loads and durable critical loads. The focus of thispaper is on the investigation of these criticed loads. Boron fibre/epoxy andgraphite fibre/epoxy materials are used in the analysis. Numericalresults indicate that, in the cases of both symmetrical and antisymmetricalply-up configuration, transient elastic critical loads and durablecriticalloads see similar trends of variation with ply angle and reach theirmaximum values at a respective ply angle of little difference regardless ofboundary conditions at two ends. It may be observed, however, that the plyangles corresponding to the peak values of these two types of critical loaddiffer by 5^\circ \sim 10^\circ in the arbitrary ply-up mode and thisdifference is dependent on the stacking sequence, the geometric parametersand the type of material as well. The conclusions drawn in the paper canexpect to be applicable to the optimal design of laminated circularcylindrical shells concerning the capability in delayedbuckling.