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This and the next two slides are from the paper:
"Optimization of composite, stiffened, imperfect panels under combined loads for service in the postbuckling regime" by David Bushnell, Computer Methods in Applied Mechanics and Engineering, Vol. 103, pp. 43-114, 1993, called "1993 PANDA2 paper".
ABSTRACT OF THE 1993 PANDA2 PAPER:
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 loadsets. The agreement of results between PANDA2 and STAGS is good enough to qualify PANDA2 as a preliminary design tool.
This is Fig. 22 from the 1993 posbuckling paper. This slide shows the PANDA2 prediction of deformation of a cross section of the locally post-buckled skin-stringer panel module for Load set 3 with axial load Nx =0, -500, -700, -800, -900, -1000, -1100 lb/in, and shear load Nxy =0, 500, 700, 800, 900, 1000,1100 lb/in. The axial compression Nx acts normal to the screen.
The local postbuckling analysis is based on the local linear bifurcation buckling mode. The postbuckling deflection pattern in the panel skin can change with increasing load above the bifurcation load because of the existence of a "flattening" unknown, "a", in the postbuckling theory and because the axial wavelength of the buckles is permitted to change as the panel is loaded further and further into its local post-buckling regime.
There are 4 unknowns in the Koiter-type nonlinear local postbuckling analysis:
"f" (the amplitude of the local buckles),
"a", (the "flattening" parameter),
"m", (the slope of the nodal lines of the buckles in the panel skin, see the next slide), and
"N", (a parameter inversely proportional to the square of the axial length of the buckles).
These four quantities change as the load is increased above the local bifurcation buckling load. The Newton method is used to solve the four simultaneous nonlinear postbuckling equations.
A sophisticated strategy of successively removing and re-introducing unknowns is used to ensure convergence of the nonlinear solution.
In this slide one can clearly see the change in the postbuckling deformation pattern as the blade stiffened panel is loaded further and further into the postbuckling region.
In the local postbuckling model it is assumed that the skin is flat. Therefore, the postbuckling model used in PANDA2 is valid only if the stringer spacing is very small compared to the radius of the cylindrical panel. (This is usually the case in optimized panels.)
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