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This and the next slide are from the 1999 paper: "Additional buckling solutions in PANDA2", by David Bushnell, Hao Jiang, and Norm F. Knight, Jr., AIAA 40th Structures, Structural Dynamics and Materials Conference, AIAA Paper 99-1233, 1999, referred to as called "1999 addsoln paper" in the next slide.
Shown here are the double trigonometric series expansions for the buckling modal displacements, u, v, w.
Local, inter-ring, and general buckling load factors and margins obtained with this "alternative" buckling model contain the string, "altsoln" in the definitions of the design margins.
The PANDA2 user can optimize panels with or without use of this "alternative" buckling model, which requires significantly more computer time than the simple PANDA-type closed form model and the semi-analytical (strip) model in which a single skin-stringer module is discretized as shown 6 slides hence.
The double-trigonometric-series "alternative" model is applied to a "patch" of the stiffened panel or shell. The size of this "patch" depends on whether the "alternative" model is being applied for local buckling, inter-ring buckling, or general buckling.
For more information about this "alternative" buckling model download the 1999 addsoln paper from this website.
ABSTRACT FROM THE 1999 PANDA2 PAPER:
Three new buckling models have been incorporated into PANDA2, a program for minimum weight design of stiffened composite panels and shells: 1. buckling of unstiffened panels or unstiffened portions of panels with use of double-trigonometric series expansions for buckling modal displacement components, u, u, w; 2. general buckling of cylindrical, stiffened panels with both rings and stringers treated as discrete beams; and 3. inter-ring buckling of cylindrical panels based on a discretized single module model containing discretized ring segments and a discretized skin-smeared-stringer cylindrical surface to which the ring is attached. Examples are provided of buckling of certain isotropic and laminated composite flat and cylindrical unstiffened and stiffened panels and shells for which the predictions from the modified PANDA2, formerly unacceptably inaccurate, are compared with predictions from STAGS, a general-purpose finite element code. The new comparisons demonstrate that the modified PANDA2 is now well qualified for preliminary design in particular cases for which it previously yielded unreliable designs and designs that were overly conservative. The optimum design of a composite ring and stringer stiffened cylindrical shell derived by PANDA2 is evaluated with use of STAGS. The optimum design of an isotropic hydrostatically compressed internally T-ring stiffened cylindrical shell optimized by PANDA2 is evaluated with use of the shell-of-revolution code BOSOR4. There is good agreement between PANDA2 predictions and STAGS and BOSOR4 predictions for buckling of the optimized designs.
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