Optimization of composite, stiffened, imperfect panels under combined loads for service in the postbuckling regime

Abstract

Local buckling and postbuckling of panels stiffened by stringers and rings and subjected to combined in-plane loads is explored with the use of a single module model that consists of one stringer and a width of panel skin equal to the stringer spacing. The cross-section of the skin-stringer module is discretized and the displacement field is assumed to vary trigonometrically in the axial direction. Local imperfections in the form of local buckling modes and overall initial axial bowing of the panel are included in the numerical model. The local postbuckling theory, based on early work of Koiter, is formulated in terms of buckling modal coefficients derived from integrals of products of the discretized displacement field and its derivatives. The principle of minimum potential energy is used to derive nonlinear algebraic equilibrium equations in terms of four unknowns, the amplitude f of the postbuckling displacement field, a postbuckling ‘flattening’ parameter a, the slope m of the nodal lines in the postbuckling displacement field, and an axial wavelength parameter N. The nonlinear equations are solved by Newton's method. An elaborate strategy is introduced in which the incidence of non-convergence is minimized by removal and re-introduction of the unknowns f, a, m, N on a one-by-one basis. This nonlinear theory has been implemented in the PANDA2 computer program, which finds minimum-weight designs of stiffened panels made of composite materials. PANDA2 is used to find the minimum weight of a cylindrical panel made of isotropic material with rectangular stringers mounted on thickened bases. The panel is optimized for three load sets, axial compression with negative axial bowing, axial compression with positive axial bowing, and combined axial compression and in-plane shear with no axial bowing. The optimum design is loaded well beyond local buckling for each load set. Critical margins of the optimized design include maximum allowable effective stress, bending-torsion buckling and general instability. The optimum design is evaluated by application of a general-purpose finite element program, STAGS, to finite element models generated by PANDA2 for each of the three load sets. The agreement of results between PANDA2 and STAGS is good enough to qualify PANDA2 as a preliminary design tool.